Select the correct answer.

Simplify the following expression:
[tex]5^{-3}[/tex]

A. [tex]5(-3)[/tex]
B. [tex]-5^3[/tex]
C. [tex]-\frac{1}{5^3}[/tex]
D. [tex]\frac{1}{5^3}[/tex]



Answer :

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]