To simplify \((32x)^ \frac{1}{5}\), we apply the rule of exponents for roots.

We can rewrite the expression as:
\[
(32x)^{\frac{1}{5}} = 32^{\frac{1}{5}} \cdot x^{\frac{1}{5}}
\]

Now, we compute \(32^{\frac{1}{5}}\). Since \(32 = 2^5\), we have:
\[
32^{\frac{1}{5}} = (2^5)^{\frac{1}{5}} = 2
\]

So, the expression simplifies to:
\[
2x^{\frac{1}{5}}
\]

The final simplified form is:
\[
2x^{\frac{1}{5}}
\]

Would you like more details or have any questions?

Here are some related questions to explore further:
1. How do we handle fractional exponents in general?
2. What happens if the base inside the parentheses was negative?
3. How does this simplify if there are coefficients like \(a \cdot 32x\)?
4. How would the expression change if it was raised to a different fractional power, like \(\frac{2}{3}\)?
5. What are the properties of exponents and roots?

**Tip**: Always simplify constants separately from variables when dealing with expressions that involve exponents.

Simplify the expression (32x)^1/5 by applying exponent rules. This detailed step-by-step guide shows how to break down and simplify the problem, using concepts such as roots and exponentiation. Learn about the power of a power rule and root simplification. Suitable for students in Grades 8-10.

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