Question

Which of the following correctly represents the expression with a rational exponent?

Answer 1

A.
`((x^7)/(z^2))^{(5)/(6)`

Step by step explanation:

We have been given an radical expression
`(\sqrt[6]{x^7z^(-2)})^5` and we are asked to choose the correct option which represent our expression with a rational exponent.

Using exponent rule for negative exponents
`a^(-m)=(1)/(a^m)` we can write our expression as:

`(\sqrt[6]{x^7*(1)/(z^(2))})^5`

`(\sqrt[6]{(x^7*1)/(z^(2))})^5`

`(\sqrt[6]{(x^7)/(z^(2))})^5`

Using the exponent property
`\sqrt[n]{a}=a^{(1)/(n)}` we can write our expression as:

`(((x^7)/(z^2))^{(1)/(6)})^5`

Using exponent property
`(a^m)^n=a^(m*n)` we can write our expression as:

`((x^7)/(z^2))^{(1)/(6)*5}`

`((x^7)/(z^2))^{(5)/(6)`

Therefore, after representing our expression with rational exponent our given expression will be:
`((x^7)/(z^2))^{(5)/(6)` and option a is the correct choice.

Answer 2

`( \sqrt[6]{x^(7) z^(-2)})^(5)=( \sqrt[6]{x^(7) (1)/(z^(2)) })^(5)= ( \sqrt[6]{ (x^(7))/(z^(2)) })^(5)= ( ( x^(7) )/(z^(2)) )^(5/6)`