Sure, let's simplify [tex]\((2^3)(2^{-4})\)[/tex] step-by-step.
1. Add the exponents and keep the same base:
When multiplying powers with the same base, you add the exponents:
[tex]\[
2^3 \cdot 2^{-4} = 2^{3 + (-4)}
\][/tex]
Simplifying the exponent:
[tex]\[
3 + (-4) = -1
\][/tex]
So, we have:
[tex]\[
2^{-1}
\][/tex]
2. Simplify the expression [tex]\(2^{-1}\)[/tex]:
An exponent of [tex]\(-1\)[/tex] means you can take the reciprocal of the base:
[tex]\[
2^{-1} = \frac{1}{2}
\][/tex]
So, the simplified expression is:
[tex]\[
2^{-1} = 0.5
\][/tex]
3. Find the reciprocal and change the sign of the exponent:
Taking the reciprocal of [tex]\(0.5\)[/tex]:
[tex]\[
\frac{1}{0.5} = 2
\][/tex]
Therefore, simplifying [tex]\((2^3)(2^{-4})\)[/tex] gives:
[tex]\[
\text{Combined Exponent: } -1
\][/tex]
[tex]\[
\text{Simplified Expression: } 0.5
\][/tex]
[tex]\[
\text{Reciprocal: } 2.0
\][/tex]
So, the final result is [tex]\(-1, 0.5, 2.0\)[/tex].
