Question

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

Answer 1

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

Answer 2

Answer:

A. $(\frac{x^7}{z^2})^{\frac{5}{6}$

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 $a^{-m}=\frac{1}{a^m}$ we can write our expression as:

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

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

$(\sqrt[6]{\frac{x^7}{z^{2}}})^5$

Using the exponent property $\sqrt[n]{a}=a^{\frac{1}{n}}$ we can write our expression as:

$((\frac{x^7}{z^2})^{\frac{1}{6}})^5$

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

$(\frac{x^7}{z^2})^{\frac{1}{6}\times5}$

$(\frac{x^7}{z^2})^{\frac{5}{6}$

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