To simplify the expression [tex]\(5^{-3}\)[/tex], let's break it down step-by-step using the properties of exponents.
### Step 1: Understand the Negative Exponent Rule
One important property of exponents is the negative exponent rule. This rule states that for any nonzero number [tex]\(a\)[/tex] and positive integer [tex]\(n\)[/tex]:
[tex]\[
a^{-n} = \frac{1}{a^n}
\][/tex]
### Step 2: Apply the Negative Exponent Rule
In our case, [tex]\(a = 5\)[/tex] and [tex]\(n = 3\)[/tex]. So, applying the negative exponent rule to [tex]\(5^{-3}\)[/tex]:
[tex]\[
5^{-3} = \frac{1}{5^3}
\][/tex]
### Step 3: Compute [tex]\(5^3\)[/tex]
Now, we need to compute the value of [tex]\(5^3\)[/tex]:
[tex]\[
5^3 = 5 \times 5 \times 5 = 125
\][/tex]
### Step 4: Substitute and Simplify
Substituting the value of [tex]\(5^3\)[/tex] back into the fraction, we get:
[tex]\[
\frac{1}{5^3} = \frac{1}{125}
\][/tex]
So, the simplified expression for [tex]\(5^{-3}\)[/tex] is [tex]\(\frac{1}{125}\)[/tex].
### Verification of Numerical Value
Let’s verify by converting [tex]\(\frac{1}{125}\)[/tex] into a decimal:
[tex]\(\frac{1}{125} = 0.008\)[/tex]
Thus, the correct answer is:
[tex]\[
\boxed{\frac{1}{5^3}}
\][/tex]
This corresponds to option A.
