To simplify the expression [tex]\( 5^{-3} \)[/tex], we need to understand the rules of exponents, particularly the rule for negative exponents.
A negative exponent indicates that we should take the reciprocal of the base and then raise it to the positive value of the exponent. Specifically, for any nonzero number [tex]\(a\)[/tex] and a positive integer [tex]\(n\)[/tex]:
[tex]\[
a^{-n} = \frac{1}{a^n}
\][/tex]
Applying this rule to our expression [tex]\( 5^{-3} \)[/tex]:
[tex]\[
5^{-3} = \frac{1}{5^3}
\][/tex]
Next, we express [tex]\(5^3\)[/tex]:
[tex]\[
5^3 = 5 \times 5 \times 5 = 125
\][/tex]
Thus,
[tex]\[
\frac{1}{5^3} = \frac{1}{125}
\][/tex]
This means the simplified form of [tex]\( 5^{-3} \)[/tex] is [tex]\( \frac{1}{5^3} \)[/tex].
Therefore, the correct answer is:
D. [tex]\( \frac{1}{5^3} \)[/tex]
