Question

Which of these choices show a pair of equivalent expression? Check all that apply.

Answer 1

Answer:

C and D

Step-by-step explanation:

using the rules of exponents

A

$125^{3/7}$ = $\sqrt[7]{125^3}$ ≠ $\sqrt[3]{125^7}$

B

($\sqrt{12}$)^7 = $12^{7/2}$ ≠ $12^{1/7}$

C

($\sqrt{4}$)^5 = $4^{5/2}$ ← correct

D

($\sqrt{8}$)^9 = $8^{9/2}$ ← correct

Answer 2

Answer:  The correct options are

(C) $4^\frac{5}{2}~~\textup{and}~~(\sqrt{4})^5.$

(D) $8^\frac{9}{2}~~\textup{and}~~(\sqrt{8})^9.$

Step-by-step explanation:  We are to select the correct pairs that shows equivalent expressions.

We will be using the following property of exponents and radicals :

$(\sqrt[b]{x})^a=x^\frac{a}{b}.$

Option (A) :

The given expressions are

$(\sqrt[3]{125})^7~~\textup{and}~~125^\frac{3}{7}.$

We have

$(\sqrt[3]{125})^7=125^\frac{7}{3}\neq 125^\frac{3}{7}.$

So, the expressions are not equivalent and option (A) is incorrect.

Option (B) :

The given expressions are

$12^\frac{1}{7}~~\textup{and}~~(\sqrt{12})^7.$

We have

$12^\frac{1}{7}=\sqrt[7]{12},\\\\(\sqrt{12})^7=12^\frac{7}{2}.$

So,

$12^\frac{1}{7}\neq (\sqrt{12})^7.$

Therefore, the expressions are not equivalent and option (B) is incorrect.

Option (C) :

The given expressions are

$4^\frac{5}{2}~~\textup{and}~~(\sqrt{4})^5.$

We have

$(\sqrt{4})^5=4^\frac{5}{2}$

Therefore, the expressions are equivalent and option (C) is correct.

Option (D) :

The given expressions are

$8^\frac{9}{2}~~\textup{and}~~(\sqrt{8})^9.$

We have

$(\sqrt{8})^9=8^\frac{9}{2}$

Therefore, the expressions are equivalent and option (D) is correct.

Thus, (C) and (D) are the correct options.