Question

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

Answer 1

The main rule we use in this exercise is


`( √(a) ) ^(m) =( a^{ (1)/(2)} )^(m) = a^{ (1)/(2)*m}=a^{ (m)/(2)}`

and the more general case:


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

A.
`( √(8) )^(9) = ( 8^{ (1)/(2)} )^(9) = 8^{ (1)/(2)*9}= 8^{ (9)/(2)}`

B.
`( √(4) ) ^(5) =( 4^{ (1)/(2)} )^(5) = 4^{ (1)/(2)*5}=4^{ (5)/(2)}`

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

D.
`( √(12) ) ^(7) =( 12^{ (1)/(2)} )^(7) = 12^{ (1)/(2)*7}=12^{ (7)/(2)}`

Answer 2

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. √(a)=a^{(1)/(2)}\\\\2.\sqrt[x]{a^y}=a^(y)/(x)`

Starting from Options

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

Option A, and Option B, are true Options.