Question

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

Answer 1

Answer with Explanation:

Two expressions are said to be equivalent, if two of them expressed in different ways, and when brought back in original form , the two expressions remain Identical.

Now, when checking out the following indices, we will keep following law of indices in mind:

$1. \sqrt{a}=a^{\frac{1}{2}}\\\\2.\sqrt[x]{a^y}=a^\frac{y}{x}$

Starting from Options

$A.8^{\frac{9}{2}}=\sqrt[2]{8^9} \\\\ B. 4^{\frac{5}{2}}=(\sqrt{4})^5}\\\\ C.(\sqrt[3]{125})^7=(125)^{\frac{7}{3}}\\\\ D. 12^{\frac{1}{7}}=\sqrt[7]{12}$

Option A, and Option B, are true Options.

Answer 2

The main rule we use in this exercise is 

$( \sqrt{a} ) ^{m} =( a^{ \frac{1}{2}} )^{m} = a^{ \frac{1}{2}*m}=a^{ \frac{m}{2}}$

and the more general case: 

$( \sqrt[n]{a} ) ^{m} =( a^{ \frac{1}{n}} )^{m} = a^{ \frac{1}{n}*m}=a^{ \frac{m}{n}}$
Using these rules, we make all the indices, or exponents, look like $a^{ \frac{m}{n}}$, and compare them to each other.

A. $( \sqrt{8} )^{9} = ( 8^{ \frac{1}{2}} )^{9} = 8^{ \frac{1}{2}*9}= 8^{ \frac{9}{2}}$

B. $( \sqrt{4} ) ^{5} =( 4^{ \frac{1}{2}} )^{5} = 4^{ \frac{1}{2}*5}=4^{ \frac{5}{2}}$

C. $( \sqrt[3]{125} ) ^{7} =( 125^{ \frac{1}{3}} )^{7} = 125^{ \frac{1}{3}*7}= 125^{ \frac{7}{3}}$

D. $( \sqrt{12} ) ^{7} =( 12^{ \frac{1}{2}} )^{7} = 12^{ \frac{1}{2}*7}=12^{ \frac{7}{2}}$