To simplify the expression [tex]\( 4^{-3} \)[/tex], follow these steps:
1. Understand the negative exponent rule:
When a base is raised to a negative exponent, such as [tex]\( a^{-n} \)[/tex], it can be rewritten as:
[tex]\[
a^{-n} = \frac{1}{a^n}
\][/tex]
Applying this rule to our expression [tex]\( 4^{-3} \)[/tex]:
[tex]\[
4^{-3} = \frac{1}{4^3}
\][/tex]
2. Compute the positive exponent:
Next, we need to calculate [tex]\( 4^3 \)[/tex]:
[tex]\[
4^3 = 4 \times 4 \times 4
\][/tex]
- First, calculate [tex]\( 4 \times 4 \)[/tex]:
[tex]\[
4 \times 4 = 16
\][/tex]
- Then, multiply the result by 4:
[tex]\[
16 \times 4 = 64
\][/tex]
Therefore, [tex]\( 4^3 = 64 \)[/tex].
3. Substitute the computed value back into the expression:
Now replace [tex]\( 4^3 \)[/tex] with 64 in the fraction:
[tex]\[
4^{-3} = \frac{1}{64}
\][/tex]
Thus, the simplified expression [tex]\( 4^{-3} \)[/tex] is:
[tex]\[
\boxed{\frac{1}{64}}
\][/tex]
