Answer :
To solve the expression [tex]\((-2)^{-3}\)[/tex], we need to understand how to deal with negative exponents. Here are the steps to simplify the expression:
1. Understand Negative Exponents:
A negative exponent means we take the reciprocal of the base raised to the opposite positive exponent. In other words, [tex]\(a^{-b} = \frac{1}{a^b}\)[/tex].
2. Apply the Rule:
Using the rule for negative exponents, we have:
[tex]\[
(-2)^{-3} = \frac{1}{(-2)^3}
\][/tex]
3. Calculate [tex]\((-2)^3\)[/tex]:
[tex]\((-2) \times (-2) \times (-2)\)[/tex] results in:
- First, multiply the first two [tex]\((-2)\)[/tex]: [tex]\((-2) \times (-2) = 4\)[/tex].
- Then multiply the result by [tex]\((-2)\)[/tex]: [tex]\(4 \times (-2) = -8\)[/tex].
4. Find the Reciprocal:
Now, we know that [tex]\( (-2)^3 = -8 \)[/tex]. Therefore:
[tex]\[
\frac{1}{(-2)^3} = \frac{1}{-8}
\][/tex]
5. Simplify:
The value [tex]\(\frac{1}{-8}\)[/tex] is equivalent to [tex]\(-\frac{1}{8}\)[/tex].
Comparing this result to the choices given:
- A. 8
- B. [tex]\(-\frac{1}{8}\)[/tex]
- C. [tex]\(\frac{1}{6}\)[/tex]
- D. 6
The correct equivalent expression for [tex]\((-2)^{-3}\)[/tex] is option B: [tex]\(-\frac{1}{8}\)[/tex].
1. Understand Negative Exponents:
A negative exponent means we take the reciprocal of the base raised to the opposite positive exponent. In other words, [tex]\(a^{-b} = \frac{1}{a^b}\)[/tex].
2. Apply the Rule:
Using the rule for negative exponents, we have:
[tex]\[
(-2)^{-3} = \frac{1}{(-2)^3}
\][/tex]
3. Calculate [tex]\((-2)^3\)[/tex]:
[tex]\((-2) \times (-2) \times (-2)\)[/tex] results in:
- First, multiply the first two [tex]\((-2)\)[/tex]: [tex]\((-2) \times (-2) = 4\)[/tex].
- Then multiply the result by [tex]\((-2)\)[/tex]: [tex]\(4 \times (-2) = -8\)[/tex].
4. Find the Reciprocal:
Now, we know that [tex]\( (-2)^3 = -8 \)[/tex]. Therefore:
[tex]\[
\frac{1}{(-2)^3} = \frac{1}{-8}
\][/tex]
5. Simplify:
The value [tex]\(\frac{1}{-8}\)[/tex] is equivalent to [tex]\(-\frac{1}{8}\)[/tex].
Comparing this result to the choices given:
- A. 8
- B. [tex]\(-\frac{1}{8}\)[/tex]
- C. [tex]\(\frac{1}{6}\)[/tex]
- D. 6
The correct equivalent expression for [tex]\((-2)^{-3}\)[/tex] is option B: [tex]\(-\frac{1}{8}\)[/tex].