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To simplify the given expression \(\left(\frac{b^3}{2a^2}\right)^5\), we apply the power of a quotient rule, which states that \(\left(\frac{x}{y}\right)^n = \frac{x^n}{y^n}\). This rule allows us to distribute the exponent \(5\) to both the numerator and the denominator.

Let's simplify step-by-step:

1. **Apply the exponent to both the numerator and the denominator:**

   \[
   \left(\frac{b^3}{2a^2}\right)^5 = \frac{(b^3)^5}{(2a^2)^5}
   \]

2. **Simplify each part separately:**

   - For the numerator: \((b^3)^5 = b^{3 \times 5} = b^{15}\)
   - For the denominator: \((2a^2)^5 = 2^5 \cdot (a^2)^5 = 32 \cdot a^{2 \times 5} = 32a^{10}\)

3. **Combine the results:**

   \[
   \frac{b^{15}}{32a^{10}}
   \]

### Final Answer:

\[
\frac{b^{15}}{32a^{10}}
\]

Would you like more details or have any questions?

Here are some related questions for further practice:

1. How do you simplify \(\left(\frac{x^2y^3}{3z^2}\right)^4\)?
2. What is the rule for raising a power to another power?
3. How do you handle negative exponents in an expression?
4. How do you distribute exponents in a fraction?
5. What is the simplified form of \(\left(\frac{4a^3b^2}{5c}\right)^3\)?

**Tip:** When simplifying expressions involving exponents, remember the laws of exponents, such as the power of a power rule \((x^m)^n = x^{mn}\).

Simplify the expression \(\left(\frac{b^3}{2a^2}\right)^5\).

Learn how to simplify the expression \(\left(\frac{b^3}{2a^2}\right)^5\) using exponent rules such as the power of a quotient and the power of a power. This problem is suitable for students in Grades 9-10.

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