Answer :
Sure! Let's break down the problem step by step to find the value of the expression:
[tex]\[
\frac{1}{5^{-5}}
\][/tex]
1. Understand the negative exponent: Recall the rule for negative exponents:
[tex]\[
a^{-n} = \frac{1}{a^n}
\][/tex]
Here, [tex]\( a = 5 \)[/tex] and [tex]\( n = 5 \)[/tex].
2. Convert the negative exponent to a positive exponent:
[tex]\[
5^{-5} = \frac{1}{5^5}
\][/tex]
3. Rewrite the original expression using the positive exponent:
[tex]\[
\frac{1}{5^{-5}} = \frac{1}{\frac{1}{5^5}}
\][/tex]
4. Simplify the complex fraction: When you divide by a fraction, it's equivalent to multiplying by its reciprocal:
[tex]\[
\frac{1}{\frac{1}{5^5}} = 5^5
\][/tex]
5. Calculate [tex]\( 5^5 \)[/tex]:
[tex]\[
5^5 = 5 \times 5 \times 5 \times 5 \times 5 = 3125
\][/tex]
So, the value of the expression [tex]\(\frac{1}{5^{-5}}\)[/tex] is:
[tex]\[
3125
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{3125}
\][/tex]
[tex]\[
\frac{1}{5^{-5}}
\][/tex]
1. Understand the negative exponent: Recall the rule for negative exponents:
[tex]\[
a^{-n} = \frac{1}{a^n}
\][/tex]
Here, [tex]\( a = 5 \)[/tex] and [tex]\( n = 5 \)[/tex].
2. Convert the negative exponent to a positive exponent:
[tex]\[
5^{-5} = \frac{1}{5^5}
\][/tex]
3. Rewrite the original expression using the positive exponent:
[tex]\[
\frac{1}{5^{-5}} = \frac{1}{\frac{1}{5^5}}
\][/tex]
4. Simplify the complex fraction: When you divide by a fraction, it's equivalent to multiplying by its reciprocal:
[tex]\[
\frac{1}{\frac{1}{5^5}} = 5^5
\][/tex]
5. Calculate [tex]\( 5^5 \)[/tex]:
[tex]\[
5^5 = 5 \times 5 \times 5 \times 5 \times 5 = 3125
\][/tex]
So, the value of the expression [tex]\(\frac{1}{5^{-5}}\)[/tex] is:
[tex]\[
3125
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{3125}
\][/tex]
