Final answer:
The expression (-32x⁻¹⁰)¹⁄₅ simplifies to -2x⁻² by taking the fifth root of both the coefficient and the variable part and combining them.
Step-by-step explanation:
To write the expression (-32x⁻¹⁰)¹⁄₅ in simplest form, we apply the property of fractional exponents, recognizing that ¹⁄₅ is the same as taking the fifth root. The expression inside the parentheses is raised to the power of ¹⁄₅, meaning both the coefficient and the variable part should be taken to the fifth root.
First, let's take the fifth root of -32, which is -2, since (-2)^5 = -32. As for the variable part, x⁻¹⁰, when we raise it to the power of ¹⁄₅, we multiply the exponents together: -10 * ¹⁄₅ = -10/5 = -2.
Therefore, the simplest form of the given expression is -2x⁻².