https://file.aiquickdraw.com/mathbot/file/1727806184221g44bb8s0.jpg

The expression shown in the image is:

\[
\frac{(2g^5)^3}{(4h^2)^3}
\]

Let's simplify this step by step:

### Step 1: Apply the exponent to both the numerator and the denominator.
For the numerator \((2g^5)^3\), apply the power of 3 to both 2 and \(g^5\):
\[
(2g^5)^3 = 2^3 \cdot (g^5)^3 = 8g^{15}
\]

For the denominator \((4h^2)^3\), apply the power of 3 to both 4 and \(h^2\):
\[
(4h^2)^3 = 4^3 \cdot (h^2)^3 = 64h^6
\]

### Step 2: Simplify the fraction:
\[
\frac{8g^{15}}{64h^6} = \frac{8}{64} \cdot \frac{g^{15}}{h^6}
\]
\[
\frac{8}{64} = \frac{1}{8}
\]

### Final Simplified Expression:
\[
\frac{g^{15}}{8h^6}
\]

This is the fully simplified form of the given expression.

Would you like any additional details or explanations on this process?

Here are some follow-up questions you might find helpful:
1. How do we simplify powers of variables with exponents?
2. What happens if the numerator and denominator are raised to different powers?
3. Can this technique apply to expressions with more complex terms?
4. How do fractional exponents work in expressions like this?
5. What are the general rules for multiplying exponents?

**Tip:** When simplifying expressions with exponents, always apply the power to both the coefficient and the variable separately before simplifying the fraction.

Simplify the expression (2g^5)^3 / (4h^2)^3.

Learn how to simplify the expression (2g^5)^3 / (4h^2)^3 step-by-step. This problem involves exponentiation and simplification, applying the laws of exponents. Suitable for Grades 9-11 students.

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