Answer :
To solve the problem of determining which expression is equivalent to [tex]\( -32^{\frac{3}{5}} \)[/tex], let's break down the steps carefully.
1. Simplify [tex]\( 32^{\frac{3}{5}} \)[/tex]:
- Recognize that [tex]\( 32 \)[/tex] can be expressed as a power of 2:
[tex]\[ 32 = 2^5 \][/tex]
- Substitute this into the exponent expression:
[tex]\[ 32^{\frac{3}{5}} = \left(2^5\right)^{\frac{3}{5}} \][/tex]
- Apply the exponent rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[ \left(2^5\right)^{\frac{3}{5}} = 2^{5 \cdot \frac{3}{5}} = 2^3 \][/tex]
- Simplify [tex]\( 2^3 \)[/tex]:
[tex]\[ 2^3 = 8 \][/tex]
2. Apply the negative sign outside the expression:
- Given that we have a negative sign outside the expression, we include it now:
[tex]\[ -32^{\frac{3}{5}} = -8 \][/tex]
3. Conclusion:
- Therefore, the expression [tex]\( -32^{\frac{3}{5}} \)[/tex] simplifies to [tex]\(-8\)[/tex].
- We match this with the given options:
[tex]\[ \text{The correct equivalent expression is: } -8 \][/tex]
Thus, the expression equivalent to [tex]\( -32^{\frac{3}{5}} \)[/tex] is [tex]\( -8 \)[/tex].
1. Simplify [tex]\( 32^{\frac{3}{5}} \)[/tex]:
- Recognize that [tex]\( 32 \)[/tex] can be expressed as a power of 2:
[tex]\[ 32 = 2^5 \][/tex]
- Substitute this into the exponent expression:
[tex]\[ 32^{\frac{3}{5}} = \left(2^5\right)^{\frac{3}{5}} \][/tex]
- Apply the exponent rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[ \left(2^5\right)^{\frac{3}{5}} = 2^{5 \cdot \frac{3}{5}} = 2^3 \][/tex]
- Simplify [tex]\( 2^3 \)[/tex]:
[tex]\[ 2^3 = 8 \][/tex]
2. Apply the negative sign outside the expression:
- Given that we have a negative sign outside the expression, we include it now:
[tex]\[ -32^{\frac{3}{5}} = -8 \][/tex]
3. Conclusion:
- Therefore, the expression [tex]\( -32^{\frac{3}{5}} \)[/tex] simplifies to [tex]\(-8\)[/tex].
- We match this with the given options:
[tex]\[ \text{The correct equivalent expression is: } -8 \][/tex]
Thus, the expression equivalent to [tex]\( -32^{\frac{3}{5}} \)[/tex] is [tex]\( -8 \)[/tex].
