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Answer :
To evaluate the numerical expression [tex]\((5^{-4})^{\frac{1}{2}}\)[/tex], we can follow these steps:
1. Evaluate [tex]\(5^{-4}\)[/tex]:
- A negative exponent means taking the reciprocal of the base raised to the positive exponent. So, [tex]\(5^{-4} = \frac{1}{5^4}\)[/tex].
- Calculate [tex]\(5^4\)[/tex]: [tex]\(5 \times 5 = 25\)[/tex], [tex]\(25 \times 5 = 125\)[/tex], and [tex]\(125 \times 5 = 625\)[/tex].
- Therefore, [tex]\(5^{-4} = \frac{1}{625}\)[/tex].
2. Take the square root:
- The expression [tex]\((5^{-4})^{\frac{1}{2}}\)[/tex] means taking the square root of [tex]\(\frac{1}{625}\)[/tex].
- The square root of [tex]\(\frac{1}{625}\)[/tex] is [tex]\(\frac{1}{\sqrt{625}}\)[/tex].
- Since [tex]\(\sqrt{625} = 25\)[/tex], we have [tex]\(\frac{1}{\sqrt{625}} = \frac{1}{25}\)[/tex].
Therefore, the value of [tex]\(\left(5^{-4}\right)^{\frac{1}{2}}\)[/tex] is [tex]\(\frac{1}{25}\)[/tex].
The correct choice from the options provided is [tex]\(\frac{1}{25}\)[/tex].
1. Evaluate [tex]\(5^{-4}\)[/tex]:
- A negative exponent means taking the reciprocal of the base raised to the positive exponent. So, [tex]\(5^{-4} = \frac{1}{5^4}\)[/tex].
- Calculate [tex]\(5^4\)[/tex]: [tex]\(5 \times 5 = 25\)[/tex], [tex]\(25 \times 5 = 125\)[/tex], and [tex]\(125 \times 5 = 625\)[/tex].
- Therefore, [tex]\(5^{-4} = \frac{1}{625}\)[/tex].
2. Take the square root:
- The expression [tex]\((5^{-4})^{\frac{1}{2}}\)[/tex] means taking the square root of [tex]\(\frac{1}{625}\)[/tex].
- The square root of [tex]\(\frac{1}{625}\)[/tex] is [tex]\(\frac{1}{\sqrt{625}}\)[/tex].
- Since [tex]\(\sqrt{625} = 25\)[/tex], we have [tex]\(\frac{1}{\sqrt{625}} = \frac{1}{25}\)[/tex].
Therefore, the value of [tex]\(\left(5^{-4}\right)^{\frac{1}{2}}\)[/tex] is [tex]\(\frac{1}{25}\)[/tex].
The correct choice from the options provided is [tex]\(\frac{1}{25}\)[/tex].
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