To simplify the expression \(\left( \frac{3y^2}{15y^6} \right)^3\), follow these steps:

### Step 1: Simplify the fraction inside the parentheses
\[
\frac{3y^2}{15y^6} = \frac{3}{15} \cdot \frac{y^2}{y^6}
\]
First, simplify the numerical fraction:
\[
\frac{3}{15} = \frac{1}{5}
\]
Now simplify the variable part:
\[
\frac{y^2}{y^6} = y^{2-6} = y^{-4}
\]
So, the expression becomes:
\[
\frac{1}{5} \cdot y^{-4} = \frac{y^{-4}}{5}
\]

### Step 2: Apply the exponent outside the parentheses
Now raise the simplified expression to the power of 3:
\[
\left( \frac{y^{-4}}{5} \right)^3 = \frac{(y^{-4})^3}{5^3}
\]
This gives:
\[
\frac{y^{-12}}{125}
\]

### Final Answer:
\[
\frac{y^{-12}}{125} = \frac{1}{125y^{12}}
\]

Would you like more details or have any questions?

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Here are five related questions you might explore:
1. How do you simplify expressions with negative exponents?
2. What happens when you multiply or divide powers with the same base?
3. How would you simplify \(\left( \frac{4x^3}{20x^5} \right)^2\)?
4. How can you handle expressions with fractions raised to a negative power?
5. What is the general rule for raising a fraction to a power?

**Tip:** When dividing powers with the same base, subtract the exponents.

Learn how to simplify the algebraic expression (3y^2/15y^6)^3 using the laws of exponents. This step-by-step solution covers key concepts in algebra, including simplifying fractions and applying exponent rules. Suitable for students in Grades 9-11.

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