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Answer :
To evaluate the expression
[tex]$$
\left(5^{-4}\right)^{\frac{1}{2}},
$$[/tex]
we start by simplifying the expression inside the parentheses.
1. The term [tex]$5^{-4}$[/tex] can be rewritten using the rule for negative exponents:
[tex]$$
5^{-4} = \frac{1}{5^4}.
$$[/tex]
2. So the expression becomes:
[tex]$$
\left(\frac{1}{5^4}\right)^{\frac{1}{2}}.
$$[/tex]
3. Taking an exponent of [tex]$\frac{1}{2}$[/tex] is equivalent to taking the square root. Therefore, we have:
[tex]$$
\sqrt{\frac{1}{5^4}} = \frac{\sqrt{1}}{\sqrt{5^4}}.
$$[/tex]
4. Since [tex]$\sqrt{1} = 1$[/tex], and [tex]$\sqrt{5^4} = 5^2$[/tex], the expression simplifies to:
[tex]$$
\frac{1}{5^2}.
$$[/tex]
5. Finally, since [tex]$5^2 = 25$[/tex], we obtain:
[tex]$$
\frac{1}{25}.
$$[/tex]
Thus, the evaluated numerical expression is
[tex]$$
\boxed{\frac{1}{25}}.
$$[/tex]
[tex]$$
\left(5^{-4}\right)^{\frac{1}{2}},
$$[/tex]
we start by simplifying the expression inside the parentheses.
1. The term [tex]$5^{-4}$[/tex] can be rewritten using the rule for negative exponents:
[tex]$$
5^{-4} = \frac{1}{5^4}.
$$[/tex]
2. So the expression becomes:
[tex]$$
\left(\frac{1}{5^4}\right)^{\frac{1}{2}}.
$$[/tex]
3. Taking an exponent of [tex]$\frac{1}{2}$[/tex] is equivalent to taking the square root. Therefore, we have:
[tex]$$
\sqrt{\frac{1}{5^4}} = \frac{\sqrt{1}}{\sqrt{5^4}}.
$$[/tex]
4. Since [tex]$\sqrt{1} = 1$[/tex], and [tex]$\sqrt{5^4} = 5^2$[/tex], the expression simplifies to:
[tex]$$
\frac{1}{5^2}.
$$[/tex]
5. Finally, since [tex]$5^2 = 25$[/tex], we obtain:
[tex]$$
\frac{1}{25}.
$$[/tex]
Thus, the evaluated numerical expression is
[tex]$$
\boxed{\frac{1}{25}}.
$$[/tex]
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