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Answer :
Sure! Let's solve the expression step-by-step:
We need to evaluate the numerical expression [tex]\(\left(5^{-4}\right)^{\frac{1}{2}}\)[/tex].
1. First, let's deal with the exponent inside the parenthesis: [tex]\(5^{-4}\)[/tex].
- The base is [tex]\(5\)[/tex].
- The exponent is [tex]\(-4\)[/tex], indicating that we need to take the reciprocal of [tex]\(5\)[/tex] raised to the 4th power.
[tex]\[
5^{-4} = \frac{1}{5^4}
\][/tex]
Now, calculate [tex]\(5^4\)[/tex]:
[tex]\[
5^4 = 5 \times 5 \times 5 \times 5 = 625
\][/tex]
So,
[tex]\[
5^{-4} = \frac{1}{625}
\][/tex]
2. Next, let's take the square root of [tex]\(\frac{1}{625}\)[/tex]:
[tex]\[
\left(\frac{1}{625}\right)^{\frac{1}{2}}
\][/tex]
The square root of [tex]\(\frac{1}{625}\)[/tex] can be written as the square root of the numerator over the square root of the denominator:
[tex]\[
\left(\frac{1}{625}\right)^{\frac{1}{2}} = \frac{1^{\frac{1}{2}}}{625^{\frac{1}{2}}}
\][/tex]
The square root of [tex]\(1\)[/tex] is:
[tex]\[
1^{\frac{1}{2}} = 1
\][/tex]
And the square root of [tex]\(625\)[/tex] is:
[tex]\[
625^{\frac{1}{2}} = 25
\][/tex]
So,
[tex]\[
\frac{1^{\frac{1}{2}}}{625^{\frac{1}{2}}} = \frac{1}{25}
\][/tex]
Thus, the value of [tex]\(\left(5^{-4}\right)^{\frac{1}{2}}\)[/tex] is [tex]\(\frac{1}{25}\)[/tex].
From the given choices, the correct answer is: [tex]\(\frac{1}{25}\)[/tex].
We need to evaluate the numerical expression [tex]\(\left(5^{-4}\right)^{\frac{1}{2}}\)[/tex].
1. First, let's deal with the exponent inside the parenthesis: [tex]\(5^{-4}\)[/tex].
- The base is [tex]\(5\)[/tex].
- The exponent is [tex]\(-4\)[/tex], indicating that we need to take the reciprocal of [tex]\(5\)[/tex] raised to the 4th power.
[tex]\[
5^{-4} = \frac{1}{5^4}
\][/tex]
Now, calculate [tex]\(5^4\)[/tex]:
[tex]\[
5^4 = 5 \times 5 \times 5 \times 5 = 625
\][/tex]
So,
[tex]\[
5^{-4} = \frac{1}{625}
\][/tex]
2. Next, let's take the square root of [tex]\(\frac{1}{625}\)[/tex]:
[tex]\[
\left(\frac{1}{625}\right)^{\frac{1}{2}}
\][/tex]
The square root of [tex]\(\frac{1}{625}\)[/tex] can be written as the square root of the numerator over the square root of the denominator:
[tex]\[
\left(\frac{1}{625}\right)^{\frac{1}{2}} = \frac{1^{\frac{1}{2}}}{625^{\frac{1}{2}}}
\][/tex]
The square root of [tex]\(1\)[/tex] is:
[tex]\[
1^{\frac{1}{2}} = 1
\][/tex]
And the square root of [tex]\(625\)[/tex] is:
[tex]\[
625^{\frac{1}{2}} = 25
\][/tex]
So,
[tex]\[
\frac{1^{\frac{1}{2}}}{625^{\frac{1}{2}}} = \frac{1}{25}
\][/tex]
Thus, the value of [tex]\(\left(5^{-4}\right)^{\frac{1}{2}}\)[/tex] is [tex]\(\frac{1}{25}\)[/tex].
From the given choices, the correct answer is: [tex]\(\frac{1}{25}\)[/tex].
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