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Evaluate the numerical expression [tex]\left(5^{-4}\right)^{\frac{1}{2}}[/tex].

A. 25
B. -25
C. [tex]\frac{1}{25}[/tex]
D. [tex]-\frac{1}{25}[/tex]

Answer :

Sure, let's break down the given expression step by step to evaluate it.

We start with the expression [tex]\(\left(5^{-4}\right)^{\frac{1}{2}}\)[/tex].

Step 1: Understand the exponentiation rules.
When you raise a power to another power, you multiply the exponents. Here, we are raising [tex]\(5^{-4}\)[/tex] to the power of [tex]\(\frac{1}{2}\)[/tex].

Step 2: Apply the exponent rule
[tex]\[
\left(5^{-4}\right)^{\frac{1}{2}} = 5^{-4 \cdot \frac{1}{2}}
\][/tex]

Step 3: Simplify the exponent multiplication
[tex]\[
-4 \cdot \frac{1}{2} = -2
\][/tex]
So,
[tex]\[
\left(5^{-4}\right)^{\frac{1}{2}} = 5^{-2}
\][/tex]

Step 4: Simplify [tex]\(5^{-2}\)[/tex]
Raising 5 to the power of -2 means [tex]\( \frac{1}{5^2} \)[/tex].

Calculate [tex]\(5^2\)[/tex]:
[tex]\[
5^2 = 25
\][/tex]

So,
[tex]\[
5^{-2} = \frac{1}{25}
\][/tex]

The correct answer is [tex]\(\frac{1}{25}\)[/tex]. Therefore, the correct option is:

[tex]\(\boxed{\frac{1}{25}}\)[/tex].

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