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The solution demonstrates how to find the value of RS by expanding the second equation, substituting the first equation, and simplifying the resulting equation.

The solution for X in the equation 2^X + 2 = 66 is 6.

To solve for a in the equation (a/2) ÷ (5/2) = 2, first rewrite the division as multiplication by the reciprocal, then multiply the numerators and denominators, simplify the fraction, isolate a by multiplying both sides by 10, and finally solve for a by dividing both sides by 2, resulting in a = 10.

To solve the equation 16/x = x^2/4, cross-multiply to get 64 = x^3, then take the cube root of both sides to find x = 4.

The solution to the equation (1/3)^(x+3) = 9^x is x = -1.

The inverse Laplace transform of 1/x^(3/2) is 2/(sqrt(pi))*sqrt(x).

The Laplace transform of sin^2(t) is 2 / (s * (s^2 + 4)).

The Laplace transform of e^(t/2) is 1/(s - 1/2), provided s > 1/2.

The equation |a - 5| = |5 - a| is proven by considering two cases: when (a - 5) is greater than or equal to zero and when (a - 5) is less than zero. In both cases, the equation holds true, demonstrating that the absolute value of the difference between a and 5 is equal to the absolute value of the difference between 5 and a for all values of a.

The rational function 1/(3x^2 + 4x + 1) is decomposed into partial fractions as (3/2)/(3x + 1) - (1/2)/(x + 1) by factoring the denominator and solving for the unknown constants using strategic values of x.