show that each of the following mathematical statement is true; a;(6 raise to 1 over 5)raise to 2 is equal to fifth root of 6 raise to 2? b;fourth root of 3 raise to -3 is equal to 1 over 27 raise to 1 over 4?

[6^(1/5)]^2 = [6^(1/5)]^2, obviously

The fifth root of 6 and 6^(1/5) are the same thing.

To prove that each of the given mathematical statements is true, we need to simplify both sides of the equation and show that they are equal. Let's start with each statement:

a) To simplify the left-hand side (LHS) of the equation:
LHS = (6^(1/5))^2

Using the property of exponents, raising a power to another power, we can multiply the exponents together:
LHS = 6^((1/5) * 2)
LHS = 6^(2/5)

Now, let's simplify the right-hand side (RHS) of the equation:
RHS = fifth root of 6^2

The fifth root of a number is equivalent to raising the number to the power of 1/n:
RHS = (6^2)^(1/5)
RHS = 6^(2/5)

Comparing the LHS and RHS, we have shown that they are equal.

b) To simplify the LHS of the equation:
LHS = (3^(1/4))^(-3)

Using the property of exponents, when a negative power is applied, it is equivalent to the reciprocal of the positive power:
LHS = (1/(3^(1/4)))^3
LHS = 1/(3^(1/4 * 3))
LHS = 1/(3^(3/4))

Now, let's simplify the RHS of the equation:
RHS = 1/(27^(1/4))

The fourth root of a number is equivalent to raising the number to the power of 1/n:
RHS = 1/(27^(1/4))
RHS = 1/(3^(3/4))

Comparing the LHS and RHS, we have shown that they are equal.

Thus, we have proven that both statements are true by simplifying each side of the equation and showing that they are equal.