To simplify the expression \(\left(\frac{2y^{5}}{3y^{2}}\right)^3\), follow these steps:

1. **Simplify the fraction inside the parentheses first:**

   \[
   \frac{2y^5}{3y^2}
   \]

   Apply the property of exponents: \(\frac{y^a}{y^b} = y^{a-b}\). Here, we have:

   \[
   \frac{y^5}{y^2} = y^{5-2} = y^3
   \]

   So, the expression becomes:

   \[
   \frac{2y^5}{3y^2} = \frac{2}{3} y^3
   \]

2. **Now raise this result to the power of 3:**

   \[
   \left( \frac{2}{3} y^3 \right)^3
   \]

   Apply the power to both the coefficient and the exponent:

   \[
   \left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3} = \frac{8}{27}
   \]

   For the exponent part, use the rule \((y^a)^b = y^{a \cdot b}\):

   \[
   (y^3)^3 = y^{3 \cdot 3} = y^9
   \]

3. **Final expression:**

   \[
   \left(\frac{2y^5}{3y^2}\right)^3 = \frac{8}{27} y^9
   \]

So, the simplified result is:

\[
\frac{8}{27} y^9
\]

Would you like further explanation on any of these steps? Or do you have other questions?

### Related Questions:
1. How do you simplify a fraction with exponents?
2. What is the rule for multiplying powers with the same base?
3. How do you raise a product to a power?
4. What is the general exponent rule for powers of products?
5. Can you simplify expressions like \(\frac{x^7}{x^4}\)?

### Tip:
When simplifying expressions with exponents, always deal with the coefficients separately and apply the exponent rules to the variables. This can make the process quicker!

Learn how to simplify the expression (2y^5 / 3y^2)^3 step by step. This problem involves key algebraic concepts such as simplifying fractions with exponents and applying exponent rules. Suitable for students in Grades 8-10.

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