To solve the expression [tex]\((-2)^{-5}\)[/tex], we need to evaluate the negative exponent. A negative exponent indicates that we should take the reciprocal of the base raised to the corresponding positive exponent. Let's break down the steps in detail:
1. Start with the base and the exponent: [tex]\((-2)^{-5}\)[/tex].
2. Recognize that a negative exponent [tex]\(-n\)[/tex] can be rewritten as the reciprocal of the positive exponent [tex]\(n\)[/tex]:
[tex]\[
a^{-n} = \frac{1}{a^n}
\][/tex]
Therefore,
[tex]\[
(-2)^{-5} = \frac{1}{(-2)^5}
\][/tex]
3. Calculate [tex]\((-2)^5\)[/tex]:
[tex]\[
(-2)^5 = (-2) \times (-2) \times (-2) \times (-2) \times (-2)
\][/tex]
- Multiplying two negative numbers gives a positive result.
- Multiplying a negative number by a positive number gives a negative result.
Continuing this way,
[tex]\[
(-2) \times (-2) = 4 \\
4 \times (-2) = -8 \\
-8 \times (-2) = 16 \\
16 \times (-2) = -32
\][/tex]
So,
[tex]\[
(-2)^5 = -32
\][/tex]
4. Substitute [tex]\((-2)^5\)[/tex] back into the reciprocal expression:
[tex]\[
(-2)^{-5} = \frac{1}{-32}
\][/tex]
5. Simplify the fraction:
[tex]\[
\frac{1}{-32} = -\frac{1}{32}
\][/tex]
6. Convert the fraction to a decimal:
[tex]\[
-\frac{1}{32} = -0.03125
\][/tex]
Therefore, the value of [tex]\((-2)^{-5}\)[/tex] is [tex]\(-0.03125\)[/tex].
Now we compare the result with the given options:
(a) [tex]\(-32\)[/tex]
Since [tex]\(-0.03125\)[/tex] does not match option (a), option (a) is incorrect.
Please make sure to check other options if provided or consider that option might be missing in the problem statement. The correct answer would be the one that matches [tex]\(-0.03125\)[/tex].
