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Answer :
To evaluate the expression [tex]\((5^{-4})^{\frac{1}{2}}\)[/tex], let's simplify it step-by-step:
1. Understand the Base Expression:
Start with the base expression inside the parentheses: [tex]\(5^{-4}\)[/tex].
- A negative exponent means to take the reciprocal. So, [tex]\(5^{-4} = \frac{1}{5^4}\)[/tex].
2. Calculate [tex]\(5^4\)[/tex]:
- [tex]\(5^4 = 5 \times 5 \times 5 \times 5 = 625\)[/tex].
- Thus, [tex]\(5^{-4} = \frac{1}{625}\)[/tex].
3. Apply the Exponent Outside the Parentheses:
Now, apply the [tex]\(\frac{1}{2}\)[/tex] exponent:
[tex]\[
\left(\frac{1}{625}\right)^{\frac{1}{2}}.
\][/tex]
- This is the same as finding the square root of [tex]\(\frac{1}{625}\)[/tex].
4. Calculate the Square Root:
- The square root of [tex]\(\frac{1}{625}\)[/tex] is [tex]\(\frac{1}{\sqrt{625}}\)[/tex].
- Since [tex]\(\sqrt{625} = 25\)[/tex], this becomes [tex]\(\frac{1}{25}\)[/tex].
Therefore, the value of the expression [tex]\((5^{-4})^{\frac{1}{2}}\)[/tex] is [tex]\(\frac{1}{25}\)[/tex].
The correct answer is [tex]\(\frac{1}{25}\)[/tex].
1. Understand the Base Expression:
Start with the base expression inside the parentheses: [tex]\(5^{-4}\)[/tex].
- A negative exponent means to take the reciprocal. So, [tex]\(5^{-4} = \frac{1}{5^4}\)[/tex].
2. Calculate [tex]\(5^4\)[/tex]:
- [tex]\(5^4 = 5 \times 5 \times 5 \times 5 = 625\)[/tex].
- Thus, [tex]\(5^{-4} = \frac{1}{625}\)[/tex].
3. Apply the Exponent Outside the Parentheses:
Now, apply the [tex]\(\frac{1}{2}\)[/tex] exponent:
[tex]\[
\left(\frac{1}{625}\right)^{\frac{1}{2}}.
\][/tex]
- This is the same as finding the square root of [tex]\(\frac{1}{625}\)[/tex].
4. Calculate the Square Root:
- The square root of [tex]\(\frac{1}{625}\)[/tex] is [tex]\(\frac{1}{\sqrt{625}}\)[/tex].
- Since [tex]\(\sqrt{625} = 25\)[/tex], this becomes [tex]\(\frac{1}{25}\)[/tex].
Therefore, the value of the expression [tex]\((5^{-4})^{\frac{1}{2}}\)[/tex] is [tex]\(\frac{1}{25}\)[/tex].
The correct answer is [tex]\(\frac{1}{25}\)[/tex].
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