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].
