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September 18, 2024

To determine if the product of $\frac{4}{5}$ and 8 will be greater than or less than 8, let's think about what it means to multiply by a fraction: * **Multiplying by a whole number:** Means you are taking groups of that number. For example, 3 x 8 means you have three groups of 8. * **Multiplying by a fraction:** Means you are taking a *part* of that number. Since the fraction $\frac{4}{5}$ is less than 1, we're taking less than a whole group of 8. **Therefore, the product of $\frac{4}{5}$ and 8 will be less than 8.** Let's calculate the answer: $\frac{4}{5} \times 8 = \frac{4 \times 8}{5} = \frac{32}{5} = 6.4$

September 14, 2024

Here's how to solve the equation \(x^{12} = (x-1)^{12}\): **1. Take the 12th root of both sides:** To get rid of the exponents, we'll take the 12th root of both sides of the equation. Remember that taking an even root introduces both positive and negative solutions: $$ \sqrt[12]{x^{12}} = \pm \sqrt[12]{(x-1)^{12}}$$ This simplifies to: $$ x = \pm (x - 1)$$ **2. Solve the two resulting equations:** Now we have two equations to solve: * **Case 1:** * x = x - 1 * 0 = -1 (This equation has no solution) * **Case 2:** * x = -(x - 1) * x = -x + 1 * 2x = 1 * x = 1/2 **Answer:** The only solution to the equation \(x^{12} = (x-1)^{12}\) is \(x = \boxed{\frac{1}{2}}\).

September 14, 2024

Here's how to solve the equation step-by-step: **1. Square both sides:** To get rid of the square roots, square both sides of the equation: $$ \left( \frac{\sqrt{A^{2}-4}}{\sqrt{A-3}} \right)^2 = 2^2 $$ This simplifies to: $$ \frac{A^2 - 4}{A - 3} = 4 $$ **2. Multiply both sides by (A-3):** $$ A^2 - 4 = 4(A - 3) $$ **3. Expand and move all terms to one side:** $$ A^2 - 4 = 4A - 12 $$ $$ A^2 - 4A + 8 = 0 $$ **4. Solve the quadratic equation:** This is a quadratic equation in the form of $ax^2 + bx + c = 0$. We can solve for *A* using the quadratic formula: $$ A = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ Where: * a = 1 * b = -4 * c = 8 Substitute the values into the quadratic formula: $$ A = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 8}}{2 \cdot 1} $$ $$ A = \frac{4 \pm \sqrt{16 - 32}}{2} $$ $$ A = \frac{4 \pm \sqrt{-16}}{2} $$ $$ A = \frac{4 \pm 4i}{2} $$ **5. Simplify:** $$ A = 2 \pm 2i $$ **Answer:** The solutions to the equation are $A = 2 + 2i$ and $A = 2 - 2i$. These are complex numbers.

September 12, 2024

## Problem Statement: Determine the maximum concentrated live load a W18×46 A992 steel beam can carry, considering bending, shear, web crippling, and local web yielding, given the following: * **Beam:** W18×46, A992 steel * **Span:** 16 ft (simply supported with lateral support at compression flange at ends only) * **Load:** Concentrated service live load * **Load Location:** 4 ft from the left support * **Bearing Length at Supports:** 4 inches * **Bearing Length at Load:** 8 inches * **Bearing Plate Thickness:** 3/4 inch * **Neglect beam weight** ## Solution: We will analyze each failure mode separately and find the minimum load capacity, which will govern the design. **1. Bending Capacity:** * **Moment Calculation:** * The maximum moment due to a concentrated load on a simply supported beam occurs at the load point. * Using the formula for the moment at a point load on a simply supported beam: * $M_{max} = \frac{P \cdot a \cdot b}{L}$ * where: * P = Concentrated load * a = Distance from left support to load = 4 ft * b = Distance from right support to load = 12 ft * L = Span of the beam = 16 ft * $M_{max} = \frac{P \cdot 4 \cdot 12}{16} = 3P$ ft-kips * **Section Modulus:** * Look up the section modulus ($S_x$) for a W18×46 beam in the AISC Steel Construction Manual. * For a W18x46, $S_x = 78.1 in^3$ * **Nominal Flexural Strength (M_n):** * $M_n = F_y \cdot S_x$ * where: * $F_y$ = Yield strength of A992 steel = 50 ksi * $M_n = 50 ksi \cdot 78.1 in^3 = 3905 in-kips = 325.42 ft-kips$ * **Design Flexural Strength (φM_n):** * $\phi_b$ = 0.9 (for flexure) * $\phi_b M_n = 0.9 \cdot 325.42 ft-kips = 292.88 ft-kips$ * **Maximum Load Capacity (P_bending):** * Equate the maximum moment due to the load to the design flexural strength: * $3P = 292.88 ft-kips$ * $P_{bending} = 97.63 kips$ **2. Shear Capacity:** * **Shear Force Calculation:** * The maximum shear force in a simply supported beam with a concentrated load occurs at the support closer to the load. * $V_{max} = \frac{P \cdot b}{L} = \frac{P \cdot 12}{16} = 0.75P$ kips * **Nominal Shear Strength (V_n):** * $V_n = 0.6 \cdot F_y \cdot A_w \cdot C_v$ * where: * $A_w$ = Area of the web = $d \cdot t_w$ * d = Depth of the beam (from AISC manual) = 18.0 in * $t_w$ = Web thickness (from AISC manual) = 0.415 in * $C_v$ = Shear coefficient (from AISC manual, typically 1.0 for beams without stiffeners) * $V_n = 0.6 \cdot 50 ksi \cdot (18.0 in \cdot 0.415 in) \cdot 1.0 = 223.8 kips$ * **Design Shear Strength (φV_n):** * $\phi_v$ = 1.0 (for shear) * $\phi_v V_n = 1.0 \cdot 223.8 kips = 223.8 kips$ * **Maximum Load Capacity (P_shear):** * Equate the maximum shear force due to the load to the design shear strength: * $0.75P = 223.8 kips$ * $P_{shear} = 298.4 kips$ **3. Web Crippling:** * **Bearing Strength at Support:** * Check the web crippling at the support with the smaller bearing length (4 inches). * $R_n = 0.75 \cdot F_y \cdot t_w^2 \cdot (1 + 3 \cdot \frac{N}{d} \cdot (\frac{t_w}{t_f})^1.5) \cdot \sqrt{\frac{F_y \cdot t_f}{t_w}}$ * where: * N = Bearing length at support = 4 in * $t_f$ = Flange thickness (from AISC manual) = 0.695 in * $R_n = 0.75 \cdot 50 ksi \cdot (0.415 in)^2 \cdot (1 + 3 \cdot \frac{4 in}{18 in} \cdot (\frac{0.415 in}{0.695 in})^1.5) \cdot \sqrt{\frac{50 ksi \cdot 0.695 in}{0.415 in}} = 104.6 kips$ * **Design Bearing Strength at Support (φR_n):** * $\phi$ = 0.75 (for web crippling) * $\phi R_n = 0.75 \cdot 104.6 kips = 78.45 kips$ * **Maximum Load Capacity at Support (P_crippling_support):** * Since the reaction at each support is half the total load: * $P_{crippling\_support} = 2 \cdot 78.45 kips = 156.9 kips$ * **Bearing Strength at Load Point:** * Repeat the above calculations for the bearing length at the load point (8 inches). * You'll find that the bearing strength at the load point is significantly higher than at the support, so it won't govern the design. **4. Local Web Yielding:** * **Bearing Yield Strength:** * $R_n = F_y \cdot t_w \cdot N = 50 ksi \cdot 0.415 in \cdot 4 in = 83 kips$ * **Design Bearing Yield Strength (φR_n):** * $\phi$ = 1.0 (for local yielding) * $\phi R_n = 1.0 \cdot 83 kips = 83 kips$ * **Maximum Load Capacity at Support (P_yielding_support):** * Since the reaction at each support is half the total load: * $P_{yielding\_support} = 2 \cdot 83 kips = 166 kips$ **5. Governing Load:** Comparing the maximum load capacities calculated for each failure mode: * $P_{bending} = 97.63 kips$ * $P_{shear} = 298.4 kips$ * $P_{crippling\_support} = 156.9 kips$ * $P_{yielding\_support} = 166 kips$ The lowest value is $P_{bending} = 97.63 kips$. ## Conclusion: The maximum concentrated live load that the W18×46 A992 steel beam can carry, considering all the specified failure modes, is **97.63 kips**. The governing failure mode is **bending**.

September 9, 2024

Let's analyze the differences between consecutive terms to find the pattern: * 5 - 2 = 3 * 9 - 5 = 4 * 14 - 9 = 5 * 20 - 14 = 6 Notice that the differences are consecutive integers: 3, 4, 5, 6... This suggests that the formula involves adding consecutive numbers to the previous term. Let's break down each term: * **2** = 1 + 1 * **5** = 2 + 3 = 1 + 1 + 3 * **9** = 5 + 4 = 1 + 1 + 3 + 4 * **14** = 9 + 5 = 1 + 1 + 3 + 4 + 5 * **20** = 14 + 6 = 1 + 1 + 3 + 4 + 5 + 6 We can see a pattern emerging: each term is the sum of consecutive integers up to a certain point. We can express this pattern with a formula. Notice that: * The first term is 1 + 1 = 1 + (1 * 2)/2 * The second term is 1 + 1 + 3 = 1 + (2 * 3)/2 * The third term is 1 + 1 + 3 + 4 = 1 + (3 * 4)/2 * And so on... This suggests the general formula: $a_n = 1 + \frac{n(n+1)}{2}$ where $a_n$ represents the nth term in the sequence. Now, let's express this set using set-builder notation: $\{a_n | a_n = 1 + \frac{n(n+1)}{2}, n \in \{1, 2, 3, 4, 5\}\}$ This reads as "the set of all $a_n$ such that $a_n$ equals 1 plus n times (n plus 1) divided by 2, where n is an element of the set {1, 2, 3, 4, 5}."

September 8, 2024

Let's break down this equation step-by-step. **(1) Expand the squares:** * The left side of the equation has two squared terms: $(r-6)^2$ and $(r-3)^2$. Remember that squaring a binomial means multiplying it by itself: * $(r-6)^2 = (r-6)(r-6)$ * $(r-3)^2 = (r-3)(r-3)$ * Now, let's expand these using the FOIL (First, Outer, Inner, Last) method: * $(r-6)(r-6) = r^2 - 6r - 6r + 36 = r^2 - 12r + 36$ * $(r-3)(r-3) = r^2 - 3r - 3r + 9 = r^2 - 6r + 9$ * Substitute these expanded forms back into the original equation: * $r^2 - 12r + 36 + r^2 - 6r + 9 = r^2$ **(2) Combine like terms:** * Combine all the $r^2$, $r$, and constant terms on the left side: * $2r^2 - 18r + 45 = r^2$ **(3) Move all terms to one side:** * Subtract $r^2$ from both sides to get a quadratic equation: * $r^2 - 18r + 45 = 0$ **(4) Solve the quadratic equation:** * We can solve this by factoring: * $(r - 15)(r - 3) = 0$ * Set each factor equal to zero and solve for *r*: * $r - 15 = 0$ or $r - 3 = 0$ * $r = 15$ or $r = 3$ **Therefore, the solutions to the equation are r = 15 and r = 3.**

September 8, 2024

Let's break down the pattern in this puzzle. * **Observation:** It seems like the numbers at the top are being combined in some way to produce the number below. * **First Example (2, 4 -> 20):** * If we multiply the two numbers: 2 * 4 = 8. This isn't 20. * If we add the two numbers and then multiply by 2: (2 + 4) * 2 = 12. This isn't 20 either. * If we square the sum of the two numbers: (2 + 4)^2 = 6^2 = 36. Still not 20. * If we add the squares of the two numbers: 2^2 + 4^2 = 4 + 16 = 20. This works! * **Second Example (3, 9 -> 90):** * Let's test our theory from the first example: 3^2 + 9^2 = 9 + 81 = 90. It works again! * **Applying the Pattern (1, 7 -> ?):** * Following the pattern, we should square the numbers and add them: 1^2 + 7^2 = 1 + 49 = 50. **Answer:** The missing number is **50 (C) **.

September 7, 2024

Let's analyze the pattern: * $8 = 56$ (8 times 7) * $7 = 42$ (7 times 6) * $6 = 30$ (6 times 5) * $5 = 20$ (5 times 4) Notice that each number on the left side is multiplied by the number that is one less than itself. Therefore: * $3 = ?$ (3 times 2) So the answer is: $3 = \boxed{6}$

September 7, 2024

The angles inside a triangle add up to $180$ degrees. Therefore, we can set up the equation: $2x + 3x + x = 180$. Combining like terms, we get: $6x = 180$. Dividing both sides by $6$, we find: $x = \boxed{30}$.

September 7, 2024

Let's break down this problem step-by-step to find the area of the blue square. **1. Visualize the Bigger Picture** * Imagine the blue square is part of a larger square formed by the outer black lines. **2. Calculate the Area of the Larger Square** * The larger square has sides of length (3 + 1) = 4 units. * The area of a square is side * side, so the larger square's area is 4 * 4 = 16 square units. **3. Calculate the Area of the Triangles** * Notice the four white triangles surrounding the blue square. Each triangle has: * A base of 3 units. * A height of 1 unit. * The area of a triangle is (1/2) * base * height. So, each triangle's area is (1/2) * 3 * 1 = 1.5 square units. **4. Subtract to Find the Blue Square's Area** * The blue square's area is the larger square's area minus the area of the four triangles. * Blue square area = 16 - (1.5 + 1.5 + 1.5 + 1.5) = 16 - 6 = 10 square units. **Answer:** The area of the colored square is **10 square units**.

September 7, 2024

Let's think step-by-step. The given figure is a parallelogram divided into four triangles. We know the area of two of these triangles: 2 m² and 5 m². **Important Fact:** The diagonals of a parallelogram divide it into four triangles of equal area. Therefore, the two remaining triangles also have areas of 2 m² and 5 m². Finally, we can calculate A + B: A = 2 m² B = 5 m² A + B = 2 m² + 5 m² = 7 m² **Answer:** b. 7 m²

September 7, 2024

Let's break down this exponential equation step-by-step. **1. Factor out a common factor:** Notice that both terms on the left-hand side of the equation have a factor of 2^T. Let's factor that out: 2^T (1 + 2¹) = 24 **2. Simplify:** Simplify the expression inside the parentheses: 2^T (1 + 2) = 24 2^T (3) = 24 **3. Isolate the exponential term:** Divide both sides of the equation by 3: 2^T = 8 **4. Express the right-hand side with the same base:** Recognize that 8 can be expressed as 2³: 2^T = 2³ **5. Solve for T:** Since the bases are the same, we can equate the exponents: T = 3 **Therefore, T = 3**

September 4, 2024

Let's call the unknown number "x". We can set up an equation to represent the problem: * **Adding to 1/2:** x + 1/2 * **Multiplying by 1/2:** x * (1/2) The problem states these two expressions are equal, so we have: x + 1/2 = x * (1/2) Now we can solve for x: 1. **Simplify:** x + 1/2 = (1/2)x 2. **Subtract (1/2)x from both sides:** (1/2)x + 1/2 = 0 3. **Subtract 1/2 from both sides:** (1/2)x = -1/2 4. **Multiply both sides by 2:** x = -1 **Therefore, the number is -1.**

September 4, 2024

Let's think step-by-step: 1. **Compare the whole number parts:** Both numbers have the same whole number part, which is 9. 2. **Compare the tenths place:** 9.11 has a 1 in the tenths place, while 9.9 has a 9 in the tenths place. Since 9 is greater than 1, 9.9 is greater than 9.11. Therefore, 9.11 is **not** greater than 9.9. Now let's find the difference: 1. **Subtract the smaller number from the larger number:** 9.9 - 9.11 2. **Perform the subtraction:** ``` 9.90 - 9.11 ------ 0.79 ``` The difference between 9.9 and 9.11 is **0.79**.

September 4, 2024

Let's approach this step-by-step: 1) First, we need to understand what information we have: - There's a triangle with one angle labeled as 25° - There's an exterior angle labeled as 150° - We're asked to find the angle labeled as X° 2) Recall the exterior angle theorem: An exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. 3) In this case, 150° is the exterior angle, and it's equal to the sum of 25° and X°. 4) We can write this as an equation: $150° = 25° + X°$ 5) To solve for X, we subtract 25° from both sides: $150° - 25° = X°$ $125° = X°$ 6) Therefore, X = 125° 7) To verify, we can check if the angles in the triangle add up to 180°: $25° + 125° + 30° = 180°$ (The 30° angle is not labeled but can be deduced) Thus, the angle X is 125°.

September 3, 2024

Let's solve this equation step by step: 1) The equation given is: $2\sqrt{x} - 14 = \frac{288}{\sqrt{x}}$ 2) Let's start by multiplying both sides by $\sqrt{x}$ to eliminate the fraction on the right side: $2x - 14\sqrt{x} = 288$ 3) Now, let's rearrange the equation to standard form: $2x - 14\sqrt{x} - 288 = 0$ 4) This is a quadratic equation in terms of $\sqrt{x}$. Let's substitute $u = \sqrt{x}$: $2u^2 - 14u - 288 = 0$ 5) Now we can solve this using the quadratic formula: $u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Where $a = 2$, $b = -14$, and $c = -288$ 6) Plugging into the quadratic formula: $u = \frac{14 \pm \sqrt{(-14)^2 - 4(2)(-288)}}{2(2)}$ 7) Simplifying: $u = \frac{14 \pm \sqrt{196 + 2304}}{4} = \frac{14 \pm \sqrt{2500}}{4} = \frac{14 \pm 50}{4}$ 8) This gives us two solutions: $u = \frac{14 + 50}{4} = 16$ or $u = \frac{14 - 50}{4} = -9$ 9) Remember that $u = \sqrt{x}$, so $x = u^2$. Since we can't have a negative square root, we only consider the positive solution: $x = 16^2 = 256$ Therefore, the solution to the equation is $x = 256$.

September 3, 2024

Let's approach this step-by-step: 1) The problem asks us to simplify $\sqrt{5 + \sqrt{11}}$. 2) This is a nested radical, where we have a square root inside another square root. 3) To simplify this, we can use the following method: Let $x = \sqrt{5 + \sqrt{11}}$ 4) Square both sides: $x^2 = 5 + \sqrt{11}$ 5) Subtract 5 from both sides: $x^2 - 5 = \sqrt{11}$ 6) Square both sides again: $(x^2 - 5)^2 = 11$ 7) Expand: $x^4 - 10x^2 + 25 = 11$ 8) Rearrange: $x^4 - 10x^2 + 14 = 0$ 9) This is a quadratic equation in $x^2$. Let $y = x^2$: $y^2 - 10y + 14 = 0$ 10) We can solve this using the quadratic formula: $y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ 11) Here, $a=1$, $b=-10$, and $c=14$ 12) Plugging into the formula: $y = \frac{10 \pm \sqrt{100 - 56}}{2} = \frac{10 \pm \sqrt{44}}{2} = \frac{10 \pm 2\sqrt{11}}{2}$ 13) Since we're looking for a positive solution (as $x$ is a square root), we take the positive root: $y = \frac{10 + 2\sqrt{11}}{2} = 5 + \sqrt{11}$ 14) Remember, $y = x^2$, so: $x = \sqrt{5 + \sqrt{11}}$ Therefore, $\sqrt{5 + \sqrt{11}}$ cannot be simplified further. It's already in its simplest form.

September 3, 2024

Let's solve this problem step by step: 1) We're given that $\frac{a}{b} = 0.25$ 2) This can be written as a fraction: $\frac{a}{b} = \frac{1}{4}$ 3) This means that $a = \frac{1}{4}b$ 4) Now, let's look at the expression we need to evaluate: $2a + 3b$ 5) We can substitute $a$ with $\frac{1}{4}b$: $2(\frac{1}{4}b) + 3b$ 6) Let's simplify the first term: $\frac{1}{2}b + 3b$ 7) Now we can add the terms: $\frac{1}{2}b + 3b = \frac{1}{2}b + \frac{6}{2}b = \frac{7}{2}b$ 8) Therefore, $2a + 3b = \frac{7}{2}b$ The final answer is $\frac{7}{2}b$. We can't simplify further because we don't know the value of $b$.

September 2, 2024

Let's approach this problem step by step, expressing each result in terms of Fibonacci terms. a) $F_1 + F_2 + F_3 + ... + F_{15}$ To solve this, we can use the formula for the sum of Fibonacci numbers up to the nth term: $\sum_{i=1}^n F_i = F_{n+2} - 1$ For $n = 15$, we get: $F_1 + F_2 + F_3 + ... + F_{15} = F_{17} - 1$ b) $F_1 + F_2 + F_3 + ... + F_{100}$ Using the same formula as above, with $n = 100$: $F_1 + F_2 + F_3 + ... + F_{100} = F_{102} - 1$ c) $F_1 + F_3 + F_5 + ... + F_{29}$ This is the sum of odd-indexed Fibonacci numbers up to $F_{29}$. There's a formula for this: $F_1 + F_3 + F_5 + ... + F_{2n-1} = F_{2n}$ Here, $2n-1 = 29$, so $n = 15$. Therefore: $F_1 + F_3 + F_5 + ... + F_{29} = F_{30}$ d) $F_2 + F_4 + F_6 + ... + F_{100}$ This is the sum of even-indexed Fibonacci numbers up to $F_{100}$. The formula for this is: $F_2 + F_4 + F_6 + ... + F_{2n} = F_{2n+1} - 1$ Here, $2n = 100$, so $n = 50$. Therefore: $F_2 + F_4 + F_6 + ... + F_{100} = F_{101} - 1$ e) $F_1 + F_4 + F_7 + ... + F_{79}$ This is the sum of Fibonacci numbers with indices that are multiples of 3 plus 1. There's no simple closed form for this sum in terms of a single Fibonacci number. f) $F_1 + F_5 + F_9 + ... + F_{97}$ Similarly, this is the sum of Fibonacci numbers with indices that are multiples of 4 plus 1. There's no simple closed form for this sum in terms of a single Fibonacci number. Final answers: a) $F_{17} - 1$ b) $F_{102} - 1$ c) $F_{30}$ d) $F_{101} - 1$ e) No simple closed form f) No simple closed form

September 2, 2024

Let's solve this step by step: 1) First, let's simplify the right side of the equation: $x^2 = (20-x) * 10/2$ $x^2 = (20-x) * 5$ 2) Now, let's expand the right side: $x^2 = 100 - 5x$ 3) Let's rearrange the equation to standard form: $x^2 + 5x - 100 = 0$ 4) This is a quadratic equation. We can solve it using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Where $a = 1$, $b = 5$, and $c = -100$ 5) Let's substitute these values into the quadratic formula: $x = \frac{-5 \pm \sqrt{5^2 - 4(1)(-100)}}{2(1)}$ 6) Simplify under the square root: $x = \frac{-5 \pm \sqrt{25 + 400}}{2} = \frac{-5 \pm \sqrt{425}}{2}$ 7) Simplify $\sqrt{425}$: $\sqrt{425} = \sqrt{25 * 17} = 5\sqrt{17}$ 8) So our solution is: $x = \frac{-5 \pm 5\sqrt{17}}{2}$ 9) We can simplify this further: $x = \frac{-5}{2} \pm \frac{5\sqrt{17}}{2}$ Therefore, the solutions are: $x = -\frac{5}{2} + \frac{5\sqrt{17}}{2}$ or $x = -\frac{5}{2} - \frac{5\sqrt{17}}{2}$ We can approximate these values: $x \approx 8.09$ or $x \approx -13.09$

September 1, 2024

Let's solve these pre-calculus problems step by step. 1. Determine the type of conic section represented by the equation: a. $x^2 - y^2 - 2x + 5y - 12 = 0$ To determine the type of conic section, we need to look at the coefficients of $x^2$ and $y^2$. - Coefficient of $x^2$ is 1 - Coefficient of $y^2$ is -1 Since these coefficients have opposite signs and equal absolute values, this equation represents a hyperbola. b. $10y^2 + 6x - 40y + 253 = 0$ In this equation: - There's no $x^2$ term - Coefficient of $y^2$ is positive (10) This equation represents a parabola. c. $2x^2 + 2y^2 + 4x - 38y + 25 = 0$ Here: - Coefficient of $x^2$ is 2 - Coefficient of $y^2$ is 2 Since these coefficients are equal and positive, this equation represents a circle. 2. Prove the identity: a. $\sin(-\theta) = -\sin \theta$ This is a fundamental trigonometric identity. It states that sine is an odd function, which means it changes sign when the angle is negated. b. $\csc^2 \theta - \cot^2 \theta = 1$ To prove this: 1) Recall the Pythagorean identity: $\sin^2 \theta + \cos^2 \theta = 1$ 2) Divide both sides by $\sin^2 \theta$: $\frac{\sin^2 \theta}{\sin^2 \theta} + \frac{\cos^2 \theta}{\sin^2 \theta} = \frac{1}{\sin^2 \theta}$ 3) Simplify: $1 + \cot^2 \theta = \csc^2 \theta$ 4) Subtract $\cot^2 \theta$ from both sides: $\csc^2 \theta - \cot^2 \theta = 1$ Thus, the identity is proved. 3. If $f(x) = 3x^2$ and $g(x) = -2x + 7$, solve the following: a. $g(3)$ $g(3) = -2(3) + 7 = -6 + 7 = 1$ b. $f(3) + g(2)$ $f(3) = 3(3^2) = 3(9) = 27$ $g(2) = -2(2) + 7 = -4 + 7 = 3$ $f(3) + g(2) = 27 + 3 = 30$ c. $[f(x)]^3$ $[f(x)]^3 = (3x^2)^3 = 27x^6$ These are the solutions to the given pre-calculus problems.

September 1, 2024

Let's solve this step by step: 1. We start with the number 1. 2. We are asked to add another 1 to it. 3. In mathematical notation, this is written as: $1 + 1$ Now, let's perform the addition: $1 + 1 = 2$ Therefore, the final answer is: $1 + 1 = 2$

August 31, 2024

Okay, let's analyze the given examples and try to produce more such examples and generalize the pattern. **Given Examples:** * $2^6 \times 2^6 = 2^{11} + 2^{11}$ * $3^8 \times 3^8 = 3^{15} + 3^{15} + 3^{15}$ **Understanding the Pattern:** Notice that in both examples: 1. We are multiplying a number raised to a power by itself (squaring it). 2. The result is the sum of the same number raised to a different power, repeated a certain number of times. Let's break down the first example: * $2^6 \times 2^6 = 2^{6+6} = 2^{12}$ (using the rule $a^m \times a^n = a^{m+n}$) * $2^{11} + 2^{11} = 2 \times 2^{11} = 2^1 \times 2^{11} = 2^{1+11} = 2^{12}$ So, $2^{12} = 2^{12}$. The pattern holds. Let's analyze the second example: * $3^8 \times 3^8 = 3^{8+8} = 3^{16}$ * $3^{15} + 3^{15} + 3^{15} = 3 \times 3^{15} = 3^1 \times 3^{15} = 3^{1+15} = 3^{16}$ Again, $3^{16} = 3^{16}$. The pattern holds. **Producing More Examples:** Let's try with base 5 and exponent 4: * $5^4 \times 5^4 = 5^{4+4} = 5^8$ * We need to find a power of 5 and a number of repetitions such that their sum equals $5^8$. Let's try $5^7$ repeated 5 times: $5^7 + 5^7 + 5^7 + 5^7 + 5^7 = 5 \times 5^7 = 5^1 \times 5^7 = 5^{1+7} = 5^8$ So, we have another example: $5^4 \times 5^4 = 5^7 + 5^7 + 5^7 + 5^7 + 5^7$ **Generalization:** Let's generalize the pattern using variables: Let the base be 'a' and the exponent be 'n'. Then, $a^n \times a^n = a^{2n}$ We need to find a power 'm' and a number of repetitions 'a' such that: $a \times a^m = a^{2n}$ $a^1 \times a^m = a^{1+m} = a^{2n}$ This implies that $1 + m = 2n$, so $m = 2n - 1$. **Generalized Form:** $a^n \times a^n = \underbrace{a^{2n-1} + a^{2n-1} + ... + a^{2n-1}}_{a \text{ times}}$ **In words:** The square of a number raised to a power 'n' is equal to the sum of that number raised to the power '2n-1', repeated 'a' (the base) times.

August 29, 2024

Let's approach this problem step by step: 1) We're given that $x = 4^{-2}$ 2) We need to find $x^x$ 3) Let's first simplify $x$: $x = 4^{-2} = \frac{1}{4^2} = \frac{1}{16}$ 4) Now, we need to calculate $(\frac{1}{16})^{\frac{1}{16}}$ 5) To solve this, we can use the property of exponents: $(a^b)^c = a^{bc}$ 6) In this case, we can rewrite our expression as: $(\frac{1}{16})^{\frac{1}{16}} = (16^{-1})^{\frac{1}{16}} = 16^{-\frac{1}{16}}$ 7) Now, let's calculate this: $16^{-\frac{1}{16}} = (2^4)^{-\frac{1}{16}} = 2^{4 \cdot (-\frac{1}{16})} = 2^{-\frac{1}{4}} = \sqrt[4]{2^{-1}} = \frac{1}{\sqrt[4]{2}}$ Therefore, $x^x = \frac{1}{\sqrt[4]{2}}$ or approximately 0.8409.

August 29, 2024

Let's approach this problem step by step: 1) First, let's recall that the sum of angles in a triangle is always 180°. 2) In the given triangle, we can see one angle is 62°. The other two angles are x and y. 3) Therefore, we can write our first equation: $x + y + 62° = 180°$ 4) Now, let's look at the angle outside the triangle. It's 128°. 5) This 128° angle forms a straight line with angle x. We know that angles on a straight line sum to 180°. 6) So we can write our second equation: $x + 128° = 180°$ 7) From this, we can find x: $x = 180° - 128° = 52°$ 8) Now that we know x, we can substitute it back into our first equation: $52° + y + 62° = 180°$ 9) Simplifying: $114° + y = 180°$ $y = 180° - 114° = 66°$ 10) Now we have both x and y. The question asks for x + y: $x + y = 52° + 66° = 118°$ Therefore, x + y = 118°.

August 28, 2024

Let's solve this step by step: 1) First, let's simplify the left side of the equation using the laws of exponents: $4^{2x-2} \times 4^{-4x+1} = 4^{(2x-2) + (-4x+1)} = 4^{-2x-1}$ 2) Now our equation looks like this: $4^{-2x-1} = \frac{1}{16}$ 3) We know that $\frac{1}{16} = 4^{-2}$ (because $4^2 = 16$) 4) So we can rewrite our equation as: $4^{-2x-1} = 4^{-2}$ 5) When the bases are the same, the exponents must be equal for the equation to hold: $-2x-1 = -2$ 6) Now we can solve for x: $-2x-1 = -2$ $-2x = -1$ $x = \frac{1}{2}$ Therefore, the solution is: $x = \frac{1}{2}$

August 28, 2024

Let's solve this problem step by step: 1) The equation we're given is: $2A^{-1} + \frac{1}{2} = 1$ 2) We need to solve for $A$. Let's start by multiplying both sides of the equation by 2: $4A^{-1} + 1 = 2$ 3) Now, let's subtract 1 from both sides: $4A^{-1} = 1$ 4) Next, we can divide both sides by 4: $A^{-1} = \frac{1}{4}$ 5) Now, to get rid of the inverse, we can take the reciprocal of both sides: $A = 4$ 6) Let's verify this solution: If $A = 4$, then $A^{-1} = \frac{1}{4}$ Plugging this back into the original equation: $2(\frac{1}{4}) + \frac{1}{2} = \frac{1}{2} + \frac{1}{2} = 1$ Therefore, the solution is: $A = 4$

August 28, 2024

Let's solve this problem step by step: 1) First, let's recall that the sum of angles in a triangle is 180°. 2) We can see that one angle of the triangle is $(50-x)°$, and another is $2x°$. 3) The third angle of the triangle is part of a straight line, which is 180°. This straight line is divided into two angles: $(5x-70)°$ and $y°$. 4) Let's write an equation based on the fact that the sum of angles in a triangle is 180°: $(50-x)° + 2x° + y° = 180°$ 5) We also know that the straight line at the base of the triangle is 180°, so: $(5x-70)° + y° = 180°$ 6) From this second equation, we can express $y$: $y = 180° - (5x-70)° = 250° - 5x$ 7) Now, let's substitute this expression for $y$ into our first equation: $(50-x)° + 2x° + (250° - 5x) = 180°$ 8) Let's solve this equation: $50 - x + 2x + 250 - 5x = 180$ $300 - 4x = 180$ $-4x = -120$ $x = 30$ 9) Now that we know $x$, we can find $y$ by substituting $x=30$ into the equation from step 6: $y = 250° - 5(30)° = 250° - 150° = 100°$ Therefore, the solution is: $x = 30$ and $y = 100$

August 27, 2024

Let's solve this problem step by step: 1) We're given that $f(x) = (x-3)^2$ 2) We're also told that $f(a) = a^2$ 3) This means that $(a-3)^2 = a^2$ 4) Let's expand the left side: $(a-3)^2 = a^2 - 6a + 9$ 5) Now we have the equation: $a^2 - 6a + 9 = a^2$ 6) Subtract $a^2$ from both sides: $-6a + 9 = 0$ 7) Add 6a to both sides: $9 = 6a$ 8) Divide both sides by 6: $\frac{9}{6} = a$ 9) Simplify: $\frac{3}{2} = a$ Therefore, $a = \frac{3}{2}$ or 1.5. To verify: $f(\frac{3}{2}) = (\frac{3}{2} - 3)^2 = (-\frac{3}{2})^2 = \frac{9}{4}$ $(\frac{3}{2})^2 = \frac{9}{4}$ Indeed, $f(a) = a^2$ when $a = \frac{3}{2}$.

August 27, 2024

Let's solve this problem step by step: 1) We are given two equations: $\frac{A + P}{A - P} = \frac{9}{2}$ $\frac{A}{P} = ?$ 2) Our goal is to find the value of $\frac{A}{P}$. 3) Let's start with the first equation. We can cross-multiply: $(A + P) \cdot 2 = (A - P) \cdot 9$ 4) Expanding this: $2A + 2P = 9A - 9P$ 5) Rearranging terms: $2A - 9A = -9P - 2P$ $-7A = -11P$ 6) Dividing both sides by -7: $A = \frac{11}{7}P$ 7) Now we have expressed A in terms of P. Let's use this to find $\frac{A}{P}$: $\frac{A}{P} = \frac{\frac{11}{7}P}{P} = \frac{11}{7}$ Therefore, the value of $\frac{A}{P}$ is $\frac{11}{7}$.

August 27, 2024

Let's solve this problem step by step to find the area of triangle ABC. 1) First, we need to recognize that this is a right-angled triangle inscribed in a circle. BC is the diameter of the circle, which means angle BAC is a right angle (90°). 2) We're given that BC = 12 cm, which is the diameter of the circle. This means the radius of the circle is 6 cm. 3) In a right-angled triangle, we can use the formula: Area = $\frac{1}{2}$ * base * height 4) We know the base (BC) is 12 cm. We need to find the height, which is the radius of the circle perpendicular to BC. 5) The radius perpendicular to BC is 6 cm (as we found in step 2). 6) Now we can apply the formula: Area = $\frac{1}{2}$ * 12 * 6 7) Let's calculate: Area = $\frac{1}{2}$ * 72 = 36 Therefore, the area of triangle ABC is 36 cm².

August 27, 2024

Let's solve this integral step by step: 1) First, let's simplify the integrand: $\frac{1}{x} \left(\frac{x}{1-x}\right) = \frac{1}{1-x}$ 2) Now our integral becomes: $\int_0^1 \frac{1}{1-x} dx$ 3) This is a standard integral form. We know that: $\int \frac{1}{1-x} dx = -\ln|1-x| + C$ 4) Now let's apply the limits: $[-\ln|1-x|]_0^1$ 5) Evaluating at the upper limit (x = 1): $\lim_{x \to 1^-} -\ln|1-x| = +\infty$ 6) Evaluating at the lower limit (x = 0): $-\ln|1-0| = -\ln(1) = 0$ 7) Therefore, the result is: $\infty - 0 = \infty$ Thus, the integral diverges to positive infinity. Final answer: The integral $\int_0^1 \frac{1}{x} \left(\frac{x}{1-x}\right) dx$ diverges.

August 26, 2024

Let's solve this problem step by step: 1) First, let's calculate how much bonus each group gets: - Total bonus: Rs 21000 - Managers get 40%: 0.40 × 21000 = Rs 8400 - Salespeople get the rest: 21000 - 8400 = Rs 12600 2) Now, let's calculate how much each person gets in the current distribution: - Each manager gets: 8400 ÷ 3 = Rs 2800 - Each salesperson gets: 12600 ÷ 7 = Rs 1800 3) If the bonus was shared equally between all 10 people: - Each person would get: 21000 ÷ 10 = Rs 2100 4) Let's compare what a salesperson gets now vs. what they would get if shared equally: - Current amount: Rs 1800 - Equal share amount: Rs 2100 5) To check if this is 25% more, we can calculate: - Percentage increase = (2100 - 1800) ÷ 1800 × 100 - = 300 ÷ 1800 × 100 - = 0.1666... × 100 - ≈ 16.67% 6) 16.67% is less than 25% Therefore, the salesperson is not correct. If the bonus was shared equally between all 10 people, they would get about 16.67% more money, not 25% more.

August 26, 2024

Let's approach this problem step by step: 1) From the first equation: $3^X \div 3^Y = 3$ This can be rewritten as: $3^{X-Y} = 3$ Therefore, $X - Y = 1$ 2) From the second equation: $3^X + 3^Y = 12$ 3) Let's use the information from step 1 to rewrite $3^X$ in terms of $3^Y$: $3^X = 3 \cdot 3^Y$ 4) Substituting this into the second equation: $3 \cdot 3^Y + 3^Y = 12$ $4 \cdot 3^Y = 12$ $3^Y = 3$ 5) Now that we know $3^Y = 3$, we can find $3^X$: $3^X = 3 \cdot 3^Y = 3 \cdot 3 = 9$ 6) The third equation asks for $3^X - 3^Y$ 7) We can now solve this: $3^X - 3^Y = 9 - 3 = 6$ Therefore, the answer to $3^X - 3^Y = ?$ is 6.

August 26, 2024

Let's solve this problem step by step: 1) First, we need to find a common denominator for all fractions. The denominators are 7, 4, 7, and 4. The least common multiple of these is 28. 2) Now, let's convert each mixed number to an improper fraction and then to an equivalent fraction with denominator 28: $15\frac{2}{7} = \frac{107}{7} = \frac{107 \times 4}{7 \times 4} = \frac{428}{28}$ $23\frac{3}{4} = \frac{95}{4} = \frac{95 \times 7}{4 \times 7} = \frac{665}{28}$ $7\frac{5}{7} = \frac{54}{7} = \frac{54 \times 4}{7 \times 4} = \frac{216}{28}$ $5\frac{1}{4} = \frac{21}{4} = \frac{21 \times 7}{4 \times 7} = \frac{147}{28}$ 3) Now we can add these fractions: $\frac{428}{28} + \frac{665}{28} + \frac{216}{28} + \frac{147}{28} = \frac{1456}{28}$ 4) Simplify this fraction: $\frac{1456}{28} = 52$ Therefore, the answer is 52. The correct option is b) 52.

August 26, 2024

Let's solve this problem step by step: 1) The equation given is: $125^{3P-4} = \sqrt{5}$ 2) First, let's simplify the right side: $\sqrt{5} = 5^{\frac{1}{2}}$ 3) Now our equation looks like: $125^{3P-4} = 5^{\frac{1}{2}}$ 4) We know that $125 = 5^3$, so let's rewrite the left side: $(5^3)^{3P-4} = 5^{\frac{1}{2}}$ 5) Using the power rule $(a^b)^c = a^{bc}$, we can simplify: $5^{3(3P-4)} = 5^{\frac{1}{2}}$ 6) Simplify the exponent on the left side: $5^{9P-12} = 5^{\frac{1}{2}}$ 7) When the bases are the same, the exponents must be equal: $9P - 12 = \frac{1}{2}$ 8) Add 12 to both sides: $9P = \frac{25}{2}$ 9) Divide both sides by 9: $P = \frac{25}{18}$ Therefore, the value of P is $\frac{25}{18}$ or approximately 1.389.

August 25, 2024

Let's solve this problem step by step: 1) First, let's simplify the numerator: $8^{-\frac{1}{2}} = \frac{1}{\sqrt{8}}$ This is because $x^{-n} = \frac{1}{x^n}$, and $\sqrt{x} = x^{\frac{1}{2}}$ 2) Now our expression looks like this: $\frac{\frac{1}{\sqrt{8}}}{\sqrt{2}} = ?$ 3) To divide fractions, we multiply by the reciprocal. So this is equivalent to: $\frac{1}{\sqrt{8}} \cdot \frac{1}{\sqrt{2}} = ?$ 4) Now we have: $\frac{1}{\sqrt{8} \cdot \sqrt{2}} = ?$ 5) Simplify under the square root: $\frac{1}{\sqrt{16}} = ?$ 6) Simplify the square root: $\frac{1}{4} = ?$ Therefore, the final answer is $\frac{1}{4}$ or 0.25.

August 25, 2024

Let's solve this equation step by step: 1) We start with the equation: $7\sqrt{3x - 5} = 28$ 2) First, let's isolate the square root term by dividing both sides by 7: $\sqrt{3x - 5} = \frac{28}{7} = 4$ 3) Now, to remove the square root, we can square both sides: $(\sqrt{3x - 5})^2 = 4^2$ $3x - 5 = 16$ 4) Add 5 to both sides to isolate the term with x: $3x = 21$ 5) Finally, divide both sides by 3 to solve for x: $x = 7$ Therefore, the solution to the equation $7\sqrt{3x - 5} = 28$ is $x = 7$. Let's verify: If $x = 7$, then $3x - 5 = 3(7) - 5 = 21 - 5 = 16$ $7\sqrt{3x - 5} = 7\sqrt{16} = 7(4) = 28$ The solution checks out.

August 25, 2024

Let's solve this problem step by step: 1) We are given three equations: $T + M = 6$ $T - M = 2$ $T^2 + M = ?$ 2) Our goal is to find the value of $T^2 + M$. 3) From the first two equations, we can solve for T and M: Adding the equations: $(T + M) + (T - M) = 6 + 2$ $2T = 8$ $T = 4$ 4) Now that we know T, we can substitute it into either of the first two equations to find M: Using $T + M = 6$: $4 + M = 6$ $M = 2$ 5) Now we have T = 4 and M = 2. 6) We can now calculate $T^2 + M$: $T^2 + M = 4^2 + 2 = 16 + 2 = 18$ Therefore, $T^2 + M = 18$.

August 25, 2024

Let's solve this equation step by step: 1) We start with the equation: $2^{x+1} - 3^x = 0$ 2) To solve this, we can move all terms to one side: $2^{x+1} = 3^x$ 3) Now, let's use the properties of exponents to simplify the left side: $2 \cdot 2^x = 3^x$ 4) We can divide both sides by $2^x$: $2 = (\frac{3}{2})^x$ 5) Now, we can take the logarithm of both sides. Let's use log base 2: $\log_2(2) = \log_2((\frac{3}{2})^x)$ 6) The left side simplifies to 1, and we can use the logarithm property on the right: $1 = x \log_2(\frac{3}{2})$ 7) Now we can solve for x: $x = \frac{1}{\log_2(\frac{3}{2})}$ 8) We can simplify this further: $x = \frac{\log_2(2)}{\log_2(3) - \log_2(2)} = \frac{1}{\log_2(3) - 1}$ Therefore, the solution is: $x = \frac{1}{\log_2(3) - 1}$ This can be approximated to about 2.7095 if a decimal answer is needed.

August 24, 2024

Let's solve this problem step-by-step using the order of operations (PEMDAS/BODMAS): 1. **Factorial:** First, we need to calculate the factorial of 3, denoted by 3! * 3! = 3 * 2 * 1 = 6 2. **Multiplication:** Next, we perform the multiplication. * 2 * 3! = 2 * 6 = 12 3. **Subtraction:** Finally, we do the subtraction. * 13 - 12 = 1 Therefore, 13 - 2 × 3! = 1

August 24, 2024

Given that $x = 7 + 4\sqrt{3}$, we want to find the value of $\sqrt{x} + \frac{1}{\sqrt{x}}$. **Step 1: Simplify $\sqrt{x}$** Notice that we can rewrite $x$ as follows: $$x = 7 + 4\sqrt{3} = 4 + 3 + 2 \cdot 2 \cdot \sqrt{3} = 2^2 + (\sqrt{3})^2 + 2 \cdot 2 \cdot \sqrt{3}$$ This looks like the expansion of a squared term: $(a + b)^2 = a^2 + b^2 + 2ab$. So, we can write: $$x = (2 + \sqrt{3})^2$$ Therefore: $$\sqrt{x} = \sqrt{(2 + \sqrt{3})^2} = 2 + \sqrt{3}$$ **Step 2: Find $\frac{1}{\sqrt{x}}$** To find $\frac{1}{\sqrt{x}}$, we'll rationalize the denominator: $$\frac{1}{\sqrt{x}} = \frac{1}{2 + \sqrt{3}} \cdot \frac{2 - \sqrt{3}}{2 - \sqrt{3}}$$ This simplifies to: $$\frac{1}{\sqrt{x}} = \frac{2 - \sqrt{3}}{(2)^2 - (\sqrt{3})^2} = \frac{2 - \sqrt{3}}{4 - 3} = 2 - \sqrt{3}$$ **Step 3: Calculate $\sqrt{x} + \frac{1}{\sqrt{x}}$** Finally, substitute the values we found: $$\sqrt{x} + \frac{1}{\sqrt{x}} = (2 + \sqrt{3}) + (2 - \sqrt{3}) = \boxed{4}$$

August 24, 2024

To solve this problem, we need to follow the order of operations (PEMDAS/BODMAS): **1. Multiplication:** 1 x 5 = 5 **2. Addition:** 5 + 5 = 10 **Therefore, the answer is B) 10.**

August 24, 2024

To solve this problem, we need to follow the order of operations (PEMDAS/BODMAS): **1. Division:** * 60 ÷ 6 = 10 **2. Multiplication:** * 10 x 2 = 20 **Therefore, the answer is 20.**

August 24, 2024

Let's solve the equation $f = \dfrac{(a-c)(b-d)}{ac-bd}$ for $c$. 1. **Multiply both sides by the denominator:** Begin by multiplying both sides of the equation by $(ac - bd)$ to get rid of the fraction: $$f(ac-bd) = (a-c)(b-d)$$ 2. **Expand both sides:** Next, expand the products on both sides of the equation: $$fac - fbd = ab - ad - bc + cd$$ 3. **Move all terms with 'c' to one side:** Collect all the terms containing 'c' on the left side and move all other terms to the right side: $$fac + bc - cd = ab - ad - fbd$$ 4. **Factor out 'c':** Factor out the common factor 'c' from the terms on the left side: $$c(fa + b - d) = ab - ad - fbd$$ 5. **Isolate 'c':** Finally, divide both sides by $(fa + b - d)$ to isolate 'c': $$c = \dfrac{ab - ad - fbd}{fa + b - d}$$ Therefore, the solution for $c$ is: $$c = \dfrac{ab - ad - fbd}{fa + b - d}$$

August 23, 2024

## Proving Bessel Functions $J_n(x)$ and $J_{n+m}(x)$ Have No Common Zeros (Except x=0) This proof relies on the following properties of Bessel functions: 1. **Bessel's differential equation:** $x^2 \frac{d^2y}{dx^2} + x \frac{dy}{dx} + (x^2 - n^2)y = 0$ where $J_n(x)$ is a solution. 2. **Recurrence relation:** $xJ_n'(x) = nJ_n(x) - xJ_{n+1}(x)$ **Proof by contradiction:** 1. **Assumption:** Let's assume that $J_n(x)$ and $J_{n+m}(x)$ have a common zero at $x = a$ where $a \neq 0$. This means: $J_n(a) = 0$ and $J_{n+m}(a) = 0$. 2. **Using the recurrence relation:** We can write the recurrence relation for $J_n(a)$ and $J_{n+m}(a)$: - $aJ_n'(a) = nJ_n(a) - aJ_{n+1}(a)$ - $aJ_{n+m}'(a) = (n+m)J_{n+m}(a) - aJ_{n+m+1}(a)$ 3. **Applying the assumption:** Since $J_n(a) = 0$ and $J_{n+m}(a) = 0$, the recurrence relations simplify to: - $aJ_n'(a) = - aJ_{n+1}(a)$ - $aJ_{n+m}'(a) = - aJ_{n+m+1}(a)$ 4. **Dividing the equations:** Dividing the first simplified recurrence relation by the second, we get: $\frac{J_n'(a)}{J_{n+m}'(a)} = \frac{J_{n+1}(a)}{J_{n+m+1}(a)}$ 5. **Repeating the process:** We can repeat steps 2-4 for $J_{n+1}(a)$ and $J_{n+m+1}(a)$, then for $J_{n+2}(a)$ and $J_{n+m+2}(a)$, and so on. This will lead to a chain of equalities: $\frac{J_n'(a)}{J_{n+m}'(a)} = \frac{J_{n+1}(a)}{J_{n+m+1}(a)} = \frac{J_{n+2}(a)}{J_{n+m+2}(a)} = ... = \frac{J_{n+m-1}(a)}{J_{n+2m-1}(a)}$ 6. **Final step:** Notice that the last term in the chain involves $J_{n+m-1}(a)$ and $J_{n+2m-1}(a)$. Since we assumed $J_{n+m}(a) = 0$, and Bessel functions have no repeated zeros (except at x=0), we know that $J_{n+m-1}(a) \neq 0$ and $J_{n+2m-1}(a) \neq 0$. This contradicts the chain of equalities we derived, meaning our initial assumption that $J_n(x)$ and $J_{n+m}(x)$ share a common zero at $x=a$ (where $a \neq 0$) must be false. **Therefore, Bessel functions $J_n(x)$ and $J_{n+m}(x)$ have no common zeros except at $x=0$.**

August 23, 2024

Let's prove this statement by contradiction. **1. Assume the opposite:** Suppose there *are* two integers, let's call them 'a' and 'b' (where a > b), that are equidistant from the square root of 2. **2. Express the assumption mathematically:** This means the distance between 'a' and √2 is equal to the distance between 'b' and √2. We can write this as: * a - √2 = √2 - b **3. Solve for √2:** * Add √2 to both sides: a = 2√2 - b * Add 'b' to both sides: a + b = 2√2 * Divide both sides by 2: (a + b) / 2 = √2 **4. The contradiction:** We've arrived at a contradiction! Here's why: * 'a' and 'b' are integers, so their sum (a + b) is also an integer. * Dividing an integer by 2 results in either an integer or a number with a decimal part of 0.5. * However, √2 is irrational. This means it cannot be expressed as a simple fraction of two integers, and its decimal representation neither terminates nor repeats. **5. Conclusion:** Our initial assumption that two integers are equidistant from √2 led to a contradiction. Therefore, the original statement is true: **No two integers are equidistant from the square root of 2.**

August 23, 2024

This is the famous Basel Problem, first posed in the 17th century and solved by Leonhard Euler. **It's not something you can prove with basic algebra or calculus.** It requires more advanced mathematical tools like Fourier series or complex analysis. Here's a brief outline of how the proof using Fourier series works: 1. **Start with the Fourier series:** The Fourier series allows you to represent a periodic function as an infinite sum of sines and cosines. 2. **Choose a specific function:** Consider the function f(x) = x^2 on the interval [-π, π], and extend it periodically to the entire real line. 3. **Find its Fourier series:** Calculate the Fourier coefficients for this function. 4. **Apply Parseval's Identity:** Parseval's identity relates the integral of a function squared to the sum of the squares of its Fourier coefficients. 5. **Evaluate at a special point:** Evaluate both sides of Parseval's identity at x = 0. This will lead to a series involving the reciprocals of squares. 6. **Manipulate the series:** Through some clever manipulations, you can transform the series into the desired form (1/1² + 1/2² + 1/3² + ...). 7. **The result:** The final result will be the equation you asked for: π²/6 = 1/1² + 1/2² + 1/3² + ... **Let me know if you'd like to see a more detailed explanation of a specific step, but be warned that a full proof can be quite involved.**

August 22, 2024

This problem delves into the realm of probability and statistics rather than a simple calculation. There's no definitive number of repeats that definitively prove a die is biased. Here's why: **Understanding Randomness** * **Chance:** Even a fair die has a 1/6 chance of landing on any side, including "6", on each individual roll. Getting multiple "6"s in a row is possible, even if unlikely. * **Sample Size:** A few rolls aren't enough to determine bias. The more times you roll, the more the results should theoretically even out if the die is fair. **Statistical Approach** To assess bias, you would use statistical tests: 1. **Hypothesis:** You start with a null hypothesis: the die is fair. 2. **Experiment:** Conduct a large number of rolls (e.g., 100, 1000). 3. **Analysis:** Analyze the frequency of each number. Statistical tests like the Chi-squared test can help determine if the observed results are significantly different from what you'd expect from a fair die. **Conclusion** You can't declare a die biased based solely on a few rolls, even if they are all the same. You need a large sample size and statistical analysis to make a strong claim about bias.

August 22, 2024

Yes, the equation $(-2 + 2) + (-2 + 2) + (-2 + 2)... = 0$ is true. Here's why: **Step 1: Simplify inside the parentheses.** * (-2 + 2) = 0 **Step 2: Substitute the simplified value back into the equation.** * 0 + 0 + 0.... = 0 **Step 3: Apply the additive identity property.** * The additive identity property states that any number plus zero equals itself. **Conclusion:** No matter how many times you add zero to itself, the sum will always be zero. Therefore, the equation holds true.

August 22, 2024

Here's how to solve this problem: **Understanding Exponents** An exponent tells you how many times to multiply a base number by itself. * For example, 2 to the power of 3 (written as 2³) means 2 * 2 * 2 = 8. **The Zero Exponent Rule** Any number (except 0) raised to the power of zero always equals 1. **Applying the Rule** In this case, we have 16 to the power of zero (16⁰). According to the zero exponent rule, this equals 1. **Answer** The correct answer is **B) 1**.

August 20, 2024

这盒牙线有50个，如果一个两块钱，那一共是： 50 个 * 2 块钱/个 = 100 块钱 所以这一盒牙线一共是 **100块钱**。

August 12, 2024

Here's how to find the derivative of the function, f(x) = 3x³ - 2x² + x - 5: **Understanding Derivatives** The derivative of a function, denoted by f'(x) or df/dx, represents the instantaneous rate of change of the function with respect to its input (x). In simpler terms, it tells us how much the function's output changes with a tiny change in the input. **Power Rule** The most common rule we use to find derivatives is the power rule. It states: * d(x^n)/dx = n*x^(n-1) This means we bring the exponent down as a coefficient and reduce the original exponent by 1. **Applying the Rules** Let's differentiate f(x) term by term: 1. **Term 1: 3x³** * Apply the power rule: (3 * 3)x^(3-1) = 9x² 2. **Term 2: -2x²** * Apply the power rule: (-2 * 2)x^(2-1) = -4x 3. **Term 3: x** * Remember that x is the same as x¹. Applying the power rule: (1 * 1)x^(1-1) = 1 4. **Term 4: -5** * The derivative of a constant (a number without any x) is always 0. **Putting it Together** Now, we combine the derivatives of each term to get the derivative of the entire function: f'(x) = 9x² - 4x + 1 **Therefore, the derivative of f(x) = 3x³ - 2x² + x - 5 is f'(x) = 9x² - 4x + 1.**

August 10, 2024

Let's break this problem down step-by-step: 1. **Find the latest bus arrival time in Bluewater:** Jeff needs to be at work at 9 am, and it takes him 8 minutes to walk from the bus station. Therefore, he needs to arrive at the Bluewater bus station no later than 8:52 am. 2. **Find the bus journey duration:** We need to consult the timetable to determine the bus journey duration from Bexleyheath to Bluewater. Unfortunately, the provided timetable excerpt does not include the arrival times at Bluewater. We need that information to proceed. 3. **Calculate the total travel time (excluding walking):** Once we have the bus journey duration, we can add it to the walking time from the Bluewater bus station to work (8 minutes). 4. **Factor in the walking time from home to the Bexleyheath bus stop:** Add the 12-minute walk from Jeff's house to the Bexleyheath bus stop to the total travel time calculated in step 3. 5. **Subtract the total travel time from the work start time:** Subtract the total travel time (including both walking segments) from 9:00 am to determine the latest time Jeff can leave his house. **Without the bus arrival times at Bluewater, we cannot determine the latest time Jeff can leave his house.**

August 10, 2024

Here's how to solve this: 1. **Find the arrival time in Paris:** The train leaving London St Pancras at 0618 arrives in Paris at 0947. 2. **Calculate the time difference:** * Hours: 09 (arrival hour) - 06 (departure hour) = 3 hours * Minutes: 47 (arrival minute) - 18 (departure minute) = 29 minutes 3. **Convert hours to minutes:** 3 hours * 60 minutes/hour = 180 minutes 4. **Add the minutes together:** 180 minutes + 29 minutes = 209 minutes **Answer:** It should take the train **209 minutes** to reach Paris.

August 10, 2024

Here's how to solve the problem step-by-step: **1. Calculate the total break time in hours:** * 20 minutes / 60 minutes/hour = 1/3 hour **2. Calculate the total driving time in hours:** * 2 1/2 hours = 2.5 hours * 85 minutes / 60 minutes/hour = 1.42 hours (approximately) * Total driving time: 2.5 hours + 1.42 hours = 3.92 hours **3. Calculate the total travel time:** * Total travel time: 3.92 hours + 1/3 hour = 4.25 hours **4. Add the total travel time to the departure time:** * 9:15 am + 4.25 hours = 1:40 pm **Therefore, Natalie arrives in Sheffield at 1:40 pm.**

August 10, 2024

Here's how to solve this problem step-by-step: **1. Convert minutes to hours and minutes:** * There are 60 minutes in an hour, so 98 minutes is equal to 1 hour and 38 minutes (98 / 60 = 1 remainder 38). **2. Add the hours to the starting time:** * 7:45 pm + 1 hour = 8:45 pm **3. Add the remaining minutes:** * 8:45 pm + 38 minutes = 9:23 pm **Therefore, the film finishes at 9:23 pm.**

August 10, 2024

Here's how to solve the problem step-by-step: **1. Calculate the time Hayley arrived at the shop:** * Hayley left home at 10:40 am. * It took her 14 minutes to walk to the shop. * Time arrived at the shop: 10:40 am + 14 minutes = 10:54 am **2. Calculate the time Hayley left the shop:** * Hayley arrived at the shop at 10:54 am. * She was at the shop for 10 minutes. * Time left the shop: 10:54 am + 10 minutes = 11:04 am **3. Calculate the time Hayley arrived at her friend's house:** * Hayley left the shop at 11:04 am. * It took her 22 minutes to walk to her friend's house. * Time arrived at friend's house: 11:04 am + 22 minutes = 11:26 am **Answer:** Hayley arrived at her friend's house at 11:26 am.

August 10, 2024

Here's how to solve this problem step-by-step: **1. Convert mixed numbers to fractions:** * 1 1/2 hours is equal to (2 * 1 + 1)/2 = 3/2 hours **2. Convert hours to minutes:** * 2 hours is equal to 2 * 60 = 120 minutes * 3/2 hours is equal to (3/2) * 60 = 90 minutes **3. Calculate the total time in minutes:** * 2 hours 25 minutes is equal to 120 minutes + 25 minutes = 145 minutes **4. Find the difference:** * The difference between the two times is 145 minutes - 90 minutes = 55 minutes **Answer:** The difference between 2 hours 25 minutes and 1 1/2 hours is **55 minutes**.

August 10, 2024

Here's how to solve the problem step-by-step: **1. Convert everything to minutes:** * **1 3/4 hours in minutes:** * 1 hour = 60 minutes * 3/4 hour = (3/4) * 60 minutes = 45 minutes * Therefore, 1 3/4 hours = 60 + 45 = 105 minutes **2. Calculate the difference:** * Difference = 105 minutes - 55 minutes = 50 minutes **Answer:** The difference between 55 minutes and 1 3/4 hours is **50 minutes**.

August 10, 2024

Here's how to convert hours to minutes: **Understanding the Conversion** * There are **60 minutes** in **1 hour**. **Calculation** 1. **Start with the given hours:** 4 hours 2. **Multiply by the conversion factor (60 minutes/hour):** 4 hours * 60 minutes/hour = 240 minutes **Answer** Therefore, 4 hours is equal to **240 minutes**.

August 10, 2024

Here's how to convert 120 minutes to hours: **Understanding the Relationship** * There are **60 minutes** in **1 hour**. **Setting up the Conversion** * We can set up a proportion to solve this: * (minutes) / (hours) = (minutes) / (hours) * We know 60 minutes equals 1 hour, and we want to find out how many hours are in 120 minutes: * 60 / 1 = 120 / x (where 'x' is the number of hours we're solving for) **Solving for x (the number of hours)** 1. **Cross-multiply:** 60 * x = 120 * 1 2. **Simplify:** 60x = 120 3. **Isolate x by dividing both sides by 60:** x = 120 / 60 4. **Calculate:** x = 2 **Answer:** 120 minutes is equal to **2 hours**.

August 6, 2024

Let's break this down: 1. **Understand the information:** We know point A is at (1,1) and line segment AB has a length of 2 units. Line segment BC has a length of 1 unit. We also see that BC is perfectly vertical. 2. **Find the x-coordinate of B and C:** Since AB is a horizontal line and A is at (1,1), point B must also have an x-coordinate of 1 + 2 = 3. Point C is directly above B, so it also has an x-coordinate of 3. 3. **Find the y-coordinate of B and C:** Since AB is horizontal, point B has the same y-coordinate as point A, which is 1. Point C is 1 unit above B, so its y-coordinate is 1 + 1 = 2. **Therefore, the coordinates of point C are (3, 2).**

August 6, 2024

Here's how to solve the problem step-by-step: **1. Find the side length of the square.** * The area of a square is side length * side length (s*s or s²) * We know the area is 9 square units, so s² = 9 * Taking the square root of both sides, we find the side length (s) = 3 units. **2. Determine the coordinates of point A.** * **Point O is at the origin (0,0).** * Since OCAB is a square, point A is 3 units to the right of point O along the x-axis and 3 units up from point O along the y-axis. **Therefore, the coordinates of point A are (3, 3).**

August 6, 2024

Let's break this problem down step-by-step. **1. Understand the Problem** We need to find the midpoint (M) of a line segment AB. We are given the coordinates of A (2,3) and B (7,3). **2. Midpoint Formula** The midpoint of a line segment is found by averaging the x-coordinates and the y-coordinates of the endpoints. The formula is: Midpoint (M) = ((x₁ + x₂)/2 , (y₁ + y₂)/2) **3. Apply the Formula** * Let A (2, 3) be (x₁, y₁) * Let B (7, 3) be (x₂, y₂) Substitute the values into the midpoint formula: M = ((2 + 7)/2 , (3 + 3)/2) **4. Calculate the Coordinates** M = (9/2 , 6/2) M = (4.5, 3) **Answer:** The coordinates of M are **(4.5, 3)**.

August 5, 2024

To find the number of real roots of the quadratic equation $x^2 + 3x + 4 = 0$, we can use the discriminant. **1. Recall the Discriminant** The discriminant, denoted as Δ (delta), is the part of the quadratic formula under the square root: Δ = b² - 4ac where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0. **2. Calculate the Discriminant** In our equation, a = 1, b = 3, and c = 4. Let's plug these values into the discriminant formula: Δ = (3)² - 4 * 1 * 4 Δ = 9 - 16 Δ = -7 **3. Interpret the Discriminant** * If Δ > 0, the equation has two distinct real roots. * If Δ = 0, the equation has one real root (a double root). * If Δ < 0, the equation has no real roots (it has two complex roots). **Conclusion** Since the discriminant Δ is -7 (which is less than 0), the equation $x^2 + 3x + 4 = 0$ has **no real roots**.

August 4, 2024

To simplify $(6x + 1)(x^2 - 3x + 4)$, we need to multiply the terms in the first parentheses by each term in the second parentheses. This is just like distributing: **Step 1: Distribute the 6x** * $6x * (x^2 - 3x + 4) = 6x^3 - 18x^2 + 24x$ **Step 2: Distribute the 1** * $1 * (x^2 - 3x + 4) = x^2 - 3x + 4$ **Step 3: Combine the results from Step 1 and Step 2** * $(6x^3 - 18x^2 + 24x) + (x^2 - 3x + 4)$ **Step 4: Combine like terms** * $6x^3 + (-18 + 1)x^2 + (24 - 3)x + 4$ **Answer:** The simplified expression is $6x^3 - 17x^2 + 21x + 4$

August 4, 2024

Let's simplify the expression step-by-step: 1. **Distribute the negative sign:** The minus sign in front of the parentheses means we multiply each term inside the second parentheses by -1: $(5x^2 + 4xy - 3) + (-1)(6x^2 + 12xy - 4y^2 + 5)$ This becomes: $5x^2 + 4xy - 3 - 6x^2 - 12xy + 4y^2 - 5$ 2. **Combine like terms:** Identify terms with the same variables and exponents: * **$x^2$ terms:** $5x^2 - 6x^2 = -x^2$ * **$xy$ terms:** $4xy - 12xy = -8xy$ * **$y^2$ terms:** $4y^2$ (no other $y^2$ terms) * **Constant terms:** $-3 - 5 = -8$ 3. **Put it all together:** Combine the simplified terms: $-x^2 - 8xy + 4y^2 - 8$ Therefore, the simplified expression is: $\boxed{-x^2 - 8xy + 4y^2 - 8}$

August 4, 2024

Given the equation $t = \sqrt{u^2 + v^2}$, let's solve for $u$ step-by-step: 1. **Isolate the square root:** Square both sides of the equation to get rid of the square root: $t^2 = u^2 + v^2$ 2. **Isolate the term with u:** Subtract $v^2$ from both sides: $t^2 - v^2 = u^2$ 3. **Solve for u:** Take the square root of both sides. Remember to consider both positive and negative solutions: $u = \pm \sqrt{t^2 - v^2}$ Therefore, the solutions for $u$ are $u = \sqrt{t^2 - v^2}$ and $u = -\sqrt{t^2 - v^2}$.

August 4, 2024

To solve this problem, we need to follow the order of operations (PEMDAS/BODMAS): 1. **Find a common denominator for all the fractions:** * The smallest common denominator for 3, 5, 4, 15, and 6 is 60. 2. **Rewrite each fraction with the common denominator:** * $\frac{1}{3} = \frac{1 \times 20}{3 \times 20} = \frac{20}{60}$ * $\frac{2}{5} = \frac{2 \times 12}{5 \times 12} = \frac{24}{60}$ * $\frac{1}{4} = \frac{1 \times 15}{4 \times 15} = \frac{15}{60}$ * $\frac{11}{15} = \frac{11 \times 4}{15 \times 4} = \frac{44}{60}$ * $\frac{13}{6} = \frac{13 \times 10}{6 \times 10} = \frac{130}{60}$ 3. **Now our problem becomes:** * $\frac{20}{60} + \frac{24}{60} + \frac{15}{60} + \frac{44}{60} - \frac{130}{60} = $ 4. **Add and subtract the numerators:** * $\frac{20 + 24 + 15 + 44 - 130}{60} = \frac{-27}{60}$ 5. **Simplify the fraction:** * $\frac{-27}{60} = \boxed{-\frac{9}{20}}$

August 4, 2024

Let's solve this problem step-by-step: 1. **Solve the operation inside the parentheses first.** We have $1\frac{1}{2} - \frac{3}{4}$. To subtract fractions, we need a common denominator. 2. **Find a common denominator.** The least common denominator for 2 and 4 is 4. We can rewrite $1\frac{1}{2}$ as $\frac{3}{2}$, and then convert it to a fraction with a denominator of 4: $\frac{3}{2} \times \frac{2}{2} = \frac{6}{4}$. 3. **Subtract the fractions:** Now we have $\frac{6}{4} - \frac{3}{4} = \frac{3}{4}$. 4. **Finally, subtract the result from 2:** $2 - \frac{3}{4}$. We can think of 2 as $\frac{8}{4}$, so the problem becomes $\frac{8}{4} - \frac{3}{4} = \frac{5}{4}$. **Answer:** $2 - \left( 1\frac{1}{2} - \frac{3}{4} \right) = \boxed{\frac{5}{4}}$ or $\boxed{1\frac{1}{4}}$.

August 4, 2024

Dividing by a fraction is the same as multiplying by the inverse of that fraction. Therefore: $\frac{1}{10} \div \frac{1}{4} = \frac{1}{10} \times \frac{4}{1}$ Now we can multiply the numerators and the denominators: $\frac{1 \times 4}{10 \times 1} = \frac{4}{10}$ Finally, we simplify the fraction: $\frac{4}{10} = \boxed{\frac{2}{5}}$

August 4, 2024

To multiply fractions, we multiply the numerators and the denominators: * **Numerators:** 5 * 6 = 30 * **Denominators:** 3 * 5 = 15 This gives us the fraction: * 30/15 We can simplify this fraction by dividing both the numerator and denominator by their greatest common factor, which is 15: * (30/15) / (15/15) = 2/1 Therefore: * $\frac{5}{3} \times \frac{6}{5} = \frac{30}{15} = \boxed{2}$

August 4, 2024

To subtract fractions, they need to have a common denominator. 1. **Find the Least Common Denominator (LCD):** The smallest common denominator for 5 and 10 is 10. 2. **Adjust the first fraction:** * Multiply the numerator and denominator of 4/5 by 2 to get a denominator of 10: (4/5) * (2/2) = 8/10 3. **Now you have:** 8/10 - 3/10 4. **Subtract the numerators:** (8 - 3)/10 = 5/10 5. **Simplify:** 5/10 simplifies to 1/2 **Answer:** $\frac{4}{5} - \frac{3}{10} = \frac{1}{2}$

August 4, 2024

To add fractions, they need to have the same denominator (the bottom number). 1. **Find the Least Common Denominator (LCD):** The LCD of 2 and 3 is 6. 2. **Rewrite the fractions with the LCD:** * $\frac{1}{2}$ is equivalent to $\frac{1 \times 3}{2 \times 3} = \frac{3}{6}$ * $\frac{1}{3}$ is equivalent to $\frac{1 \times 2}{3 \times 2} = \frac{2}{6}$ 3. **Add the fractions:** * $\frac{3}{6} + \frac{2}{6} = \frac{3+2}{6}$ 4. **Simplify:** * $\frac{5}{6}$ Therefore, $\frac{1}{2} + \frac{1}{3} = \boxed{\frac{5}{6}}$

August 4, 2024

To simplify the expression $\frac{1}{i} + \frac{1}{1+i} + \frac{1}{1-i}$, we'll need to work with the imaginary unit $i$ and manipulate these fractions. Here's a step-by-step solution: **1. Rationalize the denominators:** * For the first term, multiply the numerator and denominator by $i$: $$\frac{1}{i} \cdot \frac{i}{i} = \frac{i}{i^2}$$ Since $i^2 = -1$, we have: $$\frac{i}{-1} = -i$$ * For the second term, multiply the numerator and denominator by the conjugate of the denominator, $(1-i)$: $$\frac{1}{1+i} \cdot \frac{1-i}{1-i} = \frac{1-i}{1 - i^2}$$ Simplifying, we get: $$\frac{1-i}{1+1} = \frac{1-i}{2}$$ * For the third term, multiply the numerator and denominator by the conjugate of the denominator, $(1+i)$: $$\frac{1}{1-i} \cdot \frac{1+i}{1+i} = \frac{1+i}{1-i^2}$$ Simplifying, we get: $$\frac{1+i}{1+1} = \frac{1+i}{2}$$ **2. Combine the terms:** Now our expression looks like this: $$-i + \frac{1-i}{2} + \frac{1+i}{2}$$ Combining the fractions, we get: $$-i + \frac{1-i + 1+i}{2}$$ **3. Simplify:** Notice that the $-i$ and $+i$ terms in the numerator cancel out: $$-i + \frac{2}{2}$$ This leaves us with: $$-i + 1$$ **Therefore, the simplified form of the expression is 1 - i.**

August 4, 2024

Here's how to solve 0.5 * 0.7 step-by-step: **1. Ignore the decimal points:** Think of the problem as 5 * 7, which equals 35. **2. Count the decimal places:** * 0.5 has one decimal place. * 0.7 has one decimal place. * That's a total of two decimal places. **3. Place the decimal in the answer:** Starting from the right side of 35, move the decimal two places to the left. This gives you 0.35. **Therefore, 0.5 * 0.7 = 0.35**

August 2, 2024

Here's how to determine which number is bigger: **Understanding Decimal Places** * **Tens, Ones, Decimal Point:** Numbers to the left of the decimal point represent whole numbers (tens, ones, etc.). * **Tenths, Hundredths:** Numbers to the right of the decimal represent fractions of a whole: tenths, hundredths, thousandths, and so on. **Comparing 3.8 and 3.11** 1. **Whole Numbers:** Both numbers have the same whole number part: **3**. 2. **Tenths Place:** * 3. **8** * 3. **1** Since 8 is greater than 1, we know that 3.8 is bigger than 3.11. **Answer:** 3.8 is bigger than 3.11

August 2, 2024

Let's express the variable $b$ from the formula: 1. **Start with the formula:** $$p = \frac{a^2 + b^2 + c^2}{2}$$ 2. **Multiply both sides by 2:** $$2p = a^2 + b^2 + c^2$$ 3. **Subtract $(a^2 + c^2)$ from both sides:** $$2p - a^2 - c^2 = b^2$$ 4. **Take the square root of both sides:** $$\sqrt{2p - a^2 - c^2} = b$$ Therefore, the formula for $b$ is: $$b = \sqrt{2p - a^2 - c^2}$$

August 2, 2024

Let's solve for *x* and *y* step-by-step. **1. Expand both equations:** * Equation 1: (x - 2)(y + 5) = (x - 1)(y + 2) xy + 5x - 2y - 10 = xy + 2x - y - 2 * Equation 2: (y - 3)(x + 4) = (x + 7)(y - 4) xy + 4y - 3x - 12 = xy - 4x + 7y - 28 **2. Simplify both equations:** * Equation 1: 3x - y = 8 * Equation 2: x - 3y = -16 **3. Solve for one variable in terms of the other (let's solve for x in Equation 2):** * Equation 2: x = 3y - 16 **4. Substitute the value of x from Equation 2 into Equation 1:** * 3(3y - 16) - y = 8 **5. Simplify and solve for y:** * 9y - 48 - y = 8 * 8y = 56 * y = 7 **6. Substitute the value of y back into either Equation 1 or 2 to solve for x (let's use Equation 2):** * x = 3(7) - 16 * x = 21 - 16 * x = 5 **Answer:** * x = 5 * y = 7

August 2, 2024

Let's solve the multiplication problem step-by-step: **1. Ignore the decimal points:** Multiply the numbers as if they were whole numbers: 5 x 8 = 40 **2. Count the decimal places:** * 0.5 has one decimal place. * 0.8 has one decimal place. * Total decimal places: 1 + 1 = 2 **3. Place the decimal in the answer:** Starting from the right side of '40', move the decimal point two places to the left. Therefore, 0.5 x 0.8 = **0.40**

August 2, 2024

Let's simplify the expression: $$i^6 + i^{16} + i^{26} + i^{36} + i^{46} + i^{56}$$ **Key Idea:** The imaginary unit 'i' follows a cyclical pattern: * $i^1 = i$ * $i^2 = -1$ * $i^3 = i^2 \cdot i = -1 \cdot i = -i$ * $i^4 = (i^2)^2 = (-1)^2 = 1$ Notice that the powers of 'i' cycle through these four values. **Steps:** 1. **Divide the exponents by 4:** Focus on the remainder after dividing each exponent by 4. This is because every power of $i^4$ equals 1. * 6 divided by 4 leaves a remainder of 2. * 16 divided by 4 leaves a remainder of 0. * 26 divided by 4 leaves a remainder of 2. * 36 divided by 4 leaves a remainder of 0. * 46 divided by 4 leaves a remainder of 2. * 56 divided by 4 leaves a remainder of 0. 2. **Simplify using the cyclical pattern:** * $i^6 = i^2 = -1$ * $i^{16} = (i^4)^4 = 1^4 = 1$ * $i^{26} = i^2 = -1$ * $i^{36} = (i^4)^9 = 1^9 = 1$ * $i^{46} = i^2 = -1$ * $i^{56} = (i^4)^{14} = 1^{14} = 1$ 3. **Substitute and calculate:** Our expression becomes: (-1) + 1 + (-1) + 1 + (-1) + 1 = **0** **Therefore, the simplified form of the expression is 0.**