To solve for [tex]\(2^{-3}\)[/tex]:
1. Recognize that a negative exponent indicates the reciprocal, as [tex]\(a^{-n} = \frac{1}{a^n}\)[/tex]. Thus, [tex]\(2^{-3} = \frac{1}{2^3}\)[/tex].
2. Compute [tex]\(2^3\)[/tex]:
[tex]\[
2^3 = 2 \times 2 \times 2 = 8.
\][/tex]
3. Therefore,
[tex]\[
2^{-3} = \frac{1}{2^3} = \frac{1}{8}.
\][/tex]
Let's examine the provided choices to see which one matches [tex]\( \frac{1}{8} \)[/tex]:
A) [tex]\(-2 \times -2 \times -2 = -8\)[/tex], which is incorrect because the result is negative and not the reciprocal of 8.
B) [tex]\(2 \times -3 = -6\)[/tex], which is incorrect because it's not related to the exponentiation operation.
C) [tex]\(-3 \times -3 = 9\)[/tex], which is incorrect because exponents are not involved in this operation, and the result is positive.
D) [tex]\(\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}\)[/tex]:
[tex]\[
\frac{1}{2} \times \frac{1}{2} = \frac{1}{4},
\][/tex]
[tex]\[
\frac{1}{4} \times \frac{1}{2} = \frac{1}{8},
\][/tex]
This matches [tex]\(2^{-3} = \frac{1}{8}\)[/tex].
So, the correct choice is
D) [tex]\(\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}\)[/tex].
