📢 Shares

The solution to the equation 3^(a^2 - 10a + 25)! = 729^4 is a = 3 or a = 7, found by equating exponents after expressing both sides with the same base and solving the resulting quadratic equation.

To find the difference between ba and ab, where a and b are digits, we analyze the units and hundreds digits, leading to the conclusion that the hundreds digit of the difference is 3.

The solution to the equation 4a³ = 2⁵ is a = 2.

The equation was solved by simplifying the square root, isolating the square root term, squaring both sides, moving all terms to one side, factoring out a common factor, and solving for x, resulting in two solutions: x = 0 and x = 16/81.

The solution to the equation 3^x = 1 and 3^y = 81 is x + y = 4.

By solving the system of equations x + y = 3 and x - y = 9, we find x = 6 and y = -3. Then, substituting these values into (x/y)^2, we get 4.

The total volume of infinitely many cubes with side lengths decreasing according to the sequence 1, 1/√2, 1/√3, ... is finite, while the total surface area is infinite.

The expression 2^(n+4) - 2(2^n) / 2(2^(n+3)) simplifies to 7/8.

The derivative of xy + sin(x) = e^y with respect to x is found using implicit differentiation, applying the product rule and chain rule, and solving for dy/dx.

The derivative of x² + y² = 25 with respect to x is -x/y.