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Answer :
To evaluate the numerical expression [tex]\(\left(5^{-4}\right)^{\frac{1}{2}}\)[/tex], let's break it down step by step:
1. Understanding the expression:
- The expression inside the parentheses is [tex]\(5^{-4}\)[/tex].
- The whole expression is raised to the power of [tex]\(\frac{1}{2}\)[/tex], which is the same as taking the square root.
2. Calculate [tex]\(5^{-4}\)[/tex]:
- A negative exponent means taking the reciprocal, so [tex]\(5^{-4}\)[/tex] is equivalent to [tex]\(\frac{1}{5^4}\)[/tex].
- Calculate [tex]\(5^4\)[/tex]:
[tex]\[
5^4 = 5 \times 5 \times 5 \times 5 = 625
\][/tex]
- Therefore, [tex]\(5^{-4} = \frac{1}{625}\)[/tex].
3. Take the square root:
- Now, we need to find the square root of [tex]\(\frac{1}{625}\)[/tex].
- The square root of a fraction is the square root of the numerator divided by the square root of the denominator. In this case:
[tex]\[
\sqrt{\frac{1}{625}} = \frac{\sqrt{1}}{\sqrt{625}} = \frac{1}{25}
\][/tex]
Thus, the value of the expression [tex]\(\left(5^{-4}\right)^{\frac{1}{2}}\)[/tex] simplifies to [tex]\(\frac{1}{25}\)[/tex].
Therefore, the correct answer is: [tex]\(\frac{1}{25}\)[/tex].
1. Understanding the expression:
- The expression inside the parentheses is [tex]\(5^{-4}\)[/tex].
- The whole expression is raised to the power of [tex]\(\frac{1}{2}\)[/tex], which is the same as taking the square root.
2. Calculate [tex]\(5^{-4}\)[/tex]:
- A negative exponent means taking the reciprocal, so [tex]\(5^{-4}\)[/tex] is equivalent to [tex]\(\frac{1}{5^4}\)[/tex].
- Calculate [tex]\(5^4\)[/tex]:
[tex]\[
5^4 = 5 \times 5 \times 5 \times 5 = 625
\][/tex]
- Therefore, [tex]\(5^{-4} = \frac{1}{625}\)[/tex].
3. Take the square root:
- Now, we need to find the square root of [tex]\(\frac{1}{625}\)[/tex].
- The square root of a fraction is the square root of the numerator divided by the square root of the denominator. In this case:
[tex]\[
\sqrt{\frac{1}{625}} = \frac{\sqrt{1}}{\sqrt{625}} = \frac{1}{25}
\][/tex]
Thus, the value of the expression [tex]\(\left(5^{-4}\right)^{\frac{1}{2}}\)[/tex] simplifies to [tex]\(\frac{1}{25}\)[/tex].
Therefore, the correct answer is: [tex]\(\frac{1}{25}\)[/tex].
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