The expression shown in the image is:

$\left( \frac{-2a^4}{3b^2} \right)^3$

To simplify this expression, we apply the power of a fraction rule, which states:

$\left( \frac{A}{B} \right)^n = \frac{A^n}{B^n}$

Here, $A = -2a^4$ and $B = 3b^2$, and the exponent is 3. So we raise both the numerator and denominator to the power of 3.

Step-by-step simplification:

1. Apply the exponent to the numerator and denominator: $\left( \frac{-2a^4}{3b^2} \right)^3 = \frac{(-2a^4)^3}{(3b^2)^3}$

2. Simplify the numerator: $(-2a^4)^3 = (-2)^3 \cdot (a^4)^3 = -8a^{12}$

3. Simplify the denominator: $(3b^2)^3 = 3^3 \cdot (b^2)^3 = 27b^6$

Thus, the simplified expression is:

$\frac{-8a^{12}}{27b^6}$

Would you like more details on how the rules were applied?

Here are 5 related questions for further practice:

1. What is the result of $\left( \frac{5x^2}{4y} \right)^2$?
2. How do you simplify $\left( \frac{-3m^5}{2n^3} \right)^4$?
3. What is $\left( \frac{-7p^3}{5q^2} \right)^2$?
4. Can you explain how to distribute exponents when both the base and exponent are negative?
5. How do you handle negative exponents in expressions like $\left( \frac{-x^2}{y} \right)^{-3}$?

Tip: Always be careful with negative bases raised to an odd power—they will remain negative.