Question

Which of the following expressions is equivalent to the radical expression?

Answer 1

The expression
`\sqrt[3]{m^2n^5}` is equivalent to
`m^(2/3)n^(5/3)`.

Step-by-step explanation:

To solve this problem, we first need to understand the properties of exponents and roots. In this case, we'll be focusing on the cube root, which is represented as
`\sqrt[3]{x}`.

The general rule for reducing a radical expression of a power, such as a square root, cube root, or any other root, consists in dividing the exponent of the power by the root level. The cube root of a term is equivalent to raising that term to the power of 1/3.

Using this rule, we can rewrite the expression
`\sqrt[3]{m^2n^5}` as
`m^{(2)/(3)}n^{(5)/(3)}`.

Now, let's compare this expression with the choices given in the question:

A.
`m^{(3)/(2)}n^{(3)/(5)}` is incorrect, as it does not match our calculated expression.

B.
`m^{(2)/(3)}n^{(5)/(3)}` is a match with our calculated expression, so this is the correct choice.

C.
`m^(5)n^(8)` is incorrect, as the exponents are much too high.

D.
`mn^2` is incorrect, as this simplifies to
`m^1n^2`, which does not match our calculated expression.

So the correct answer is (B)
`m^{(2)/(3)}n^{(5)/(3)}`.

Answer 2

The correct answer is B)
`m^(2/3)n^(5/3)`.

The denominator of the exponent is the root we are taking. The numerator is the exponent of the radicand. This means 3 will be the denominator, since it is a cubed root, and 2 will be the numerator of m while 5 will be the numerator of n.