To determine which expression is equivalent to [tex]\( -32^{\frac{3}{5}} \)[/tex], we can break down the steps of solving the expression [tex]\( -32^{\frac{3}{5}} \)[/tex].
1. Interpret the base and the exponent:
- The base is 32.
- The exponent is [tex]\(\frac{3}{5}\)[/tex].
2. Rewrite the base using its prime factors:
- [tex]\( 32 = 2^5 \)[/tex].
3. Apply the exponent to the base using properties of exponents:
- [tex]\( (2^5)^{\frac{3}{5}} \)[/tex].
- Using the power rule [tex]\( (a^m)^n = a^{mn} \)[/tex], we get:
[tex]\[
(2^5)^{\frac{3}{5}} = 2^{5 \cdot \frac{3}{5}} = 2^3 = 8.
\][/tex]
4. Apply the negative sign outside the power:
- Hence, [tex]\( -32^{\frac{3}{5}} = -8 \)[/tex].
Thus, the expression equivalent to [tex]\( -32^{\frac{3}{5}} \)[/tex] is [tex]\(-8\)[/tex].
To confirm, we observe the given multiple choices:
- [tex]\(-8\)[/tex]
- [tex]\( -\sqrt[3]{32^5} \)[/tex]
- [tex]\( \frac{1}{\sqrt[3]{32^5}} \)[/tex]
- [tex]\( \frac{1}{8} \)[/tex]
We can immediately match our result with the first option.
So the expression equivalent to [tex]\( -32^{\frac{3}{5}} \)[/tex] is [tex]\(\boxed{-8}\)[/tex].
