Question

Which of these choices show a pair of equivalent expressions? A.7^5/7 and (√ 7)^5 B.6^7/2 and (√ 6)^7 C.(4√ 81)^7 and 81^7/4 D.5^2/3 and (√ 5)^3

Answer 1

We will use the following law of indices (or 'index law') to check each pair of expression

$x^{ \frac{m}{n}} = ( \sqrt[n]{x} )^{m}$

With fractional power, the denominator is the root and the numerator is the power of the term. When the denominator is 2, we usually only write the normal square symbol (√). Denominator other than 2, we usually write the value of the root, for example, the cubic root ∛

Option A - Incorrect
$7^{ \frac{5}{7}}$ should equal to $( \sqrt[7]{7} )^{5}$

Option B - Correct
$6^{ \frac{7}{2} }$ does equal to $( \sqrt{6}) ^{7}$

Option C - Incorrect
$4( \sqrt{81} )^{7}$ should equal to $(4)( 81^{ \frac{7}{2} } )$

Option D - Incorrect
$5^{ \frac{2}{3} }$ should equal to $( \sqrt[3]{5} )^{2}$

Answer 2

By rewriting the exponents, we will see that the only pair of equivalent expressions are pair B.

6^(7/2) and (√6)^7

How to find equivalent expressions?

Here we need to remember the properties:

√x = x^(1/2)

(x^n)^m = x^(n*m)

Now let's apply that to the given pairs of expressions to see which ones are equivalent.

A) 7^(5/7) and (√ 7)^5

We can rewrite the second expression as:

(√7)^5 = (7^(1/2))^5 = 7^(5/2)

Then the expressions are not equivalent.

B) 6^(7/2) and (√6)^7

We can rewrite the second expression as:

(√6)^7 =  (6^(1/2))^7 = 6^(1/2).

So the expressions are equivalent.

C) (4√ 81)^7 and 81^(7/4)

The first expression can be rewritten as:

(4√ 81)^7 = (4^7)*(81^(7/2))

Clearly, these expressions are not equivalent.

D) 5^(2/3) and (√ 5)^3

The second expression can be rewritten as:

5^(3/2).

So these are not equivalent

Then the only pair of expressions that are equivalent is pair B.

If you want to learn more about exponents, you can read:

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