To determine which expression is equivalent to [tex]\(-32^{\frac{3}{5}}\)[/tex], let's analyze the options step by step.
Given:
[tex]\[ -32^{\frac{3}{5}} \][/tex]
We have four options to check against our expression.
### Option A: [tex]\(-8\)[/tex]
This option suggests that [tex]\( -32^{\frac{3}{5}} \)[/tex] is equal to [tex]\(-8\)[/tex].
### Option B: [tex]\(-\sqrt[3]{32^5}\)[/tex]
To analyze this, we need to understand what [tex]\( -\sqrt[3]{32^5} \)[/tex] means:
[tex]\[ -\sqrt[3]{32^5} = - \left( 32^5 \right)^{\frac{1}{3}} \][/tex]
[tex]\[ = - \left( 32^{\frac{5}{3}} \right) \][/tex]
### Option C: [tex]\(\frac{1}{\sqrt[3]{32^5}}\)[/tex]
We simplify this option as well:
[tex]\[ \frac{1}{\sqrt[3]{32^5}} = \left( 32^5 \right)^{-\frac{1}{3}} \][/tex]
[tex]\[ = 32^{-\frac{5}{3}} \][/tex]
[tex]\[ = \left( 32^{\frac{5}{3}} \right)^{-1} \][/tex]
[tex]\[ = \frac{1}{32^{\frac{5}{3}}} \][/tex]
### Option D: [tex]\(\frac{1}{8}\)[/tex]
This option is straightforward, it is the reciprocal of 8:
[tex]\[ \frac{1}{8} \][/tex]
### Numerical Evaluation
Given that [tex]\( -32^{\frac{3}{5}} \approx -7.999999999999999 \)[/tex], we would compare this value with our given options:
1. Option A states that the expression is [tex]\(-8\)[/tex]. The numerical value [tex]\(-7.999999999999999\)[/tex] is indeed very close to [tex]\(-8\)[/tex].
2. Checking Option B:
[tex]\[ -\sqrt[3]{32^5} = -\left( 32^{5/3} \right) \approx -8. This does not match -7.999999999999999. \][/tex]
3. Checking Option C:
[tex]\[ \frac{1}{\sqrt[3]{32^5}} = \frac{1}{\left( 32^{5/3} \right)} \approx \frac{1}{-8} \approx - \frac{1}{8}. This does not match -7.999999999999999. \][/tex]
4. Option D's value [tex]\(\frac{1}{8} \approx 0.125\)[/tex], is quite different from the numerical value -7.999999999999999.
Given this analysis and evaluating all options, the expression [tex]\(-8\)[/tex] is indeed equivalent to the expression [tex]\(-32^{\frac{3}{5}}\)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{-8} \][/tex]
