To solve the expression \(2^{-5}\), let's break it down step-by-step:
1. Understanding the Negative Exponent:
The expression \(2^{-5}\) involves a base of 2 raised to a negative exponent, -5. A negative exponent indicates a reciprocal. Specifically, \(2^{-5} = \frac{1}{2^5}\).
2. Calculating the Positive Exponent Term:
Next, we need to determine what \(2^5\) equals. This involves multiplying 2 by itself 5 times:
[tex]\[
2^5 = 2 \times 2 \times 2 \times 2 \times 2
\][/tex]
[tex]\[
2 \times 2 = 4
\][/tex]
[tex]\[
4 \times 2 = 8
\][/tex]
[tex]\[
8 \times 2 = 16
\][/tex]
[tex]\[
16 \times 2 = 32
\][/tex]
3. Substituting Back to the Reciprocal Form:
Now that we know \(2^5 = 32\), we can substitute this into our original reciprocal expression:
[tex]\[
2^{-5} = \frac{1}{2^5} = \frac{1}{32}
\][/tex]
So, filling in the blanks in the given expression:
[tex]\[
2^{-5} = \frac{1}{2 \cdot 2^4} = \frac{1}{32}
\][/tex]
Thus,
[tex]\[
2^{-5} = \frac{1}{32}
\][/tex]
Therefore, the answer is:
[tex]\[
2^{-5} = 0.03125 \quad \text{and} \quad 2^5 = 32
\][/tex]
The solution shows the value of both the negative and positive exponentiation:
[tex]\[
2^{-5} = 0.03125 \\
2^5 = 32
\][/tex]
