To determine which of the given expressions is equivalent to [tex]\( -32^{\frac{3}{5}} \)[/tex], we can compare each option to this value.
First, let's evaluate the expression [tex]\( -32^{\frac{3}{5}} \)[/tex]:
1. Option 1: [tex]\(-8\)[/tex]
- We need to determine if [tex]\( \-8 \)[/tex] is equivalent to [tex]\( -32^{\frac{3}{5}} \)[/tex].
2. Option 2: [tex]\(-\sqrt[3]{32^5}\)[/tex]
- This expression can be simplified as follows:
[tex]\[
-\sqrt[3]{32^5} = - ( 32^{5/3} )
\][/tex]
- We need to determine if this value matches with [tex]\(-32^{\frac{3}{5}}\)[/tex].
3. Option 3: [tex]\(\frac{1}{\sqrt[3]{32^5}}\)[/tex]
- This expression can be simplified as follows:
[tex]\[
\frac{1}{\sqrt[3]{32^5}} = \frac{1}{ 32^{5/3} }
\][/tex]
- We need to determine if this value matches with [tex]\(-32^{\frac{3}{5}}\)[/tex].
4. Option 4: [tex]\(\frac{1}{8}\)[/tex]
- This expression simplifies as [tex]\(\frac{1}{8}\)[/tex].
- We need to determine if this value matches with [tex]\(-32^{\frac{3}{5}}\)[/tex].
Next, let's look at the numerical results of these comparisons:
- The difference between [tex]\( -32^{\frac{3}{5}} \)[/tex] and [tex]\(-8\)[/tex] is approximately [tex]\(-8.881784197001252e-16\)[/tex]. This value is extremely close to zero, indicating that [tex]\( -32^{\frac{3}{5}} \approx -8 \)[/tex].
- The difference between [tex]\( -32^{\frac{3}{5}} \)[/tex] and [tex]\(-\sqrt[3]{32^5}\)[/tex] is roughly [tex]\(-314.5397887730874\)[/tex]. This large value suggests that [tex]\( -32^{\frac{3}{5}} \)[/tex] is not equivalent to [tex]\(-\sqrt[3]{32^5}\)[/tex].
- The difference between [tex]\( -32^{\frac{3}{5}} \)[/tex] and [tex]\(\frac{1}{\sqrt[3]{32^5}}\)[/tex] is approximately [tex]\(-8.003100392679624\)[/tex]. This indicates they are not the same.
- The difference between [tex]\( -32^{\frac{3}{5}} \)[/tex] and [tex]\(\frac{1}{8}\)[/tex] is approximately 8.125, again indicating they are not the same.
Based on these numerical comparisons, the expression [tex]\( -32^{\frac{3}{5}} \)[/tex] is closest to [tex]\(-8\)[/tex]. Therefore, the correct equivalent expression is:
[tex]\[
-32^{\frac{3}{5}} \approx -8
\][/tex]
Thus, the correct answer is:
[tex]\[
-8
\][/tex]
