https://file.aiquickdraw.com/mathbot/file/1727115580377n2ra8hup.jpg

Let's first analyze the expression given in the image:

The original expression is:

\[
\left( \frac{1}{\sqrt[3]{y}} \right)^{\frac{-1}{5}}
\]

### Step 1: Simplify the expression
We can use the rule of exponents:

\[
\left( \frac{1}{a} \right)^b = a^{-b}
\]

So:

\[
\left( \frac{1}{\sqrt[3]{y}} \right)^{\frac{-1}{5}} = \left( y^{-\frac{1}{3}} \right)^{\frac{-1}{5}}
\]

### Step 2: Apply the power of a power rule
The power rule states:

\[
\left( a^m \right)^n = a^{m \cdot n}
\]

So:

\[
\left( y^{-\frac{1}{3}} \right)^{\frac{-1}{5}} = y^{\left( -\frac{1}{3} \cdot -\frac{1}{5} \right)} = y^{\frac{1}{15}}
\]

Thus, the simplified expression is:

\[
y^{\frac{1}{15}}
\]

### Step 3: Match with the options
The correct option that corresponds to \( y^{\frac{1}{15}} \) is the fourth option in the image.

Would you like more details or explanations? Here are some additional related questions for practice:

1. How would you simplify \(\left( y^{2/3} \right)^{1/4}\)?
2. What is the rule for negative exponents?
3. How do you express a cube root in terms of fractional exponents?
4. How would you simplify \(\left( \frac{1}{y^{1/2}} \right)^{-3}\)?
5. What is the general rule for applying exponents to roots?

**Tip:** When working with fractional exponents, remember that the denominator represents the root, and the numerator represents the power.

Which expression is equivalent to (1 / ∛y )^(-1/5)?

Learn how to simplify the expression (1 / ∛y )^(-1/5) using exponent rules and the power of a power rule. This problem covers key algebraic concepts suitable for students in Grades 9-11.

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