To simplify the expression [tex]\((2^3)(2^{-4})\)[/tex] correctly, follow these detailed steps:
1. Add the Exponents:
When multiplying two powers with the same base, you add their exponents.
[tex]\[
2^3 \cdot 2^{-4} = 2^{3 + (-4)} = 2^{-1}
\][/tex]
So, the expression simplifies to [tex]\(2^{-1}\)[/tex].
2. Understand the Simplified Expression:
The expression [tex]\(2^{-1}\)[/tex] represents a base of 2 raised to the power of -1.
3. Find the Reciprocal and Change the Sign of the Exponent:
Since negative exponents indicate reciprocals, [tex]\(2^{-1}\)[/tex] is the same as [tex]\(\frac{1}{2}\)[/tex].
[tex]\[
2^{-1} = \frac{1}{2}
\][/tex]
Now let’s summarize:
- The sum of the exponents [tex]\(3 + (-4)\)[/tex] is [tex]\(-1\)[/tex].
- [tex]\(2^{-1}\)[/tex] simplifies to [tex]\(\frac{1}{2}\)[/tex].
Hence, each calculated component is:
- Exponent Sum: [tex]\(-1\)[/tex]
- Simplified Expression: [tex]\(2^{-1} = \frac{1}{2}\)[/tex]
- Reciprocal of the Simplified Expression: 2 (which is [tex]\(\frac{1}{\frac{1}{2}} = 2\)[/tex])
Therefore, the final results are:
- Exponent Sum: [tex]\(-1\)[/tex]
- Value of Simplified Expression: [tex]\(0.5\)[/tex]
- Reciprocal: [tex]\(2.0\)[/tex]
