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Answer :
Sure, let's evaluate the expression \(\left(5^{-4}\right)^{\frac{1}{2}}\) step by step.
1. Evaluate the inner exponentiation \(5^{-4}\):
- Since the exponent is negative, \(5^{-4}\) can be rewritten as \(\frac{1}{5^4}\).
- Now calculate \(5^4\):
[tex]\[
5^4 = 5 \times 5 \times 5 \times 5 = 625
\][/tex]
- So, \(5^{-4} = \frac{1}{625}\).
2. Evaluate the outer exponentiation \(\left(\frac{1}{625}\right)^{\frac{1}{2}}\):
- The exponent \(\frac{1}{2}\) signifies taking the square root.
- We need to find the square root of \(\frac{1}{625}\).
[tex]\[
\sqrt{\frac{1}{625}} = \frac{\sqrt{1}}{\sqrt{625}} = \frac{1}{\sqrt{625}}
\][/tex]
- Calculate \(\sqrt{625}\):
[tex]\[
\sqrt{625} = 25
\][/tex]
- Therefore, \(\frac{1}{\sqrt{625}} = \frac{1}{25}\).
After evaluating the numerical expression step by step, we find that \(\left(5^{-4}\right)^{\frac{1}{2}} = \frac{1}{25}\).
So, the correct answer is [tex]\(\frac{1}{25}\)[/tex].
1. Evaluate the inner exponentiation \(5^{-4}\):
- Since the exponent is negative, \(5^{-4}\) can be rewritten as \(\frac{1}{5^4}\).
- Now calculate \(5^4\):
[tex]\[
5^4 = 5 \times 5 \times 5 \times 5 = 625
\][/tex]
- So, \(5^{-4} = \frac{1}{625}\).
2. Evaluate the outer exponentiation \(\left(\frac{1}{625}\right)^{\frac{1}{2}}\):
- The exponent \(\frac{1}{2}\) signifies taking the square root.
- We need to find the square root of \(\frac{1}{625}\).
[tex]\[
\sqrt{\frac{1}{625}} = \frac{\sqrt{1}}{\sqrt{625}} = \frac{1}{\sqrt{625}}
\][/tex]
- Calculate \(\sqrt{625}\):
[tex]\[
\sqrt{625} = 25
\][/tex]
- Therefore, \(\frac{1}{\sqrt{625}} = \frac{1}{25}\).
After evaluating the numerical expression step by step, we find that \(\left(5^{-4}\right)^{\frac{1}{2}} = \frac{1}{25}\).
So, the correct answer is [tex]\(\frac{1}{25}\)[/tex].
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