The expression provided in the image is:

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

To simplify this, follow these steps:

Step 1: Simplify the fraction inside the parentheses.

We have:

$\frac{2a^2b^3c^4}{a^3b^2c^3}$

Simplify each variable:

• For $a$, subtract the exponents: $a^{2-3} = a^{-1}$.
• For $b$, subtract the exponents: $b^{3-2} = b^1 = b$.
• For $c$, subtract the exponents: $c^{4-3} = c^1 = c$.

So the expression becomes:

$2a^{-1}bc$

Step 2: Apply the square to the entire simplified expression.

We need to square $2a^{-1}bc$:

$(2a^{-1}bc)^2 = 2^2a^{-2}(b^2)(c^2)$

This simplifies to:

$4a^{-2}b^2c^2$

Final Answer:

The simplified form of the given expression is:

$4 \frac{b^2c^2}{a^2}$

Let me know if you have any questions or if you'd like further clarification! Here are five related questions to expand your understanding:

1. How would you simplify an expression with different variables and exponents?
2. Can you simplify $\left( \frac{3x^4y^3}{x^2y^5} \right)^3$?
3. What happens if there is a negative exponent in the denominator instead of the numerator?
4. How do you simplify expressions with more than three variables?
5. What are the rules for adding and subtracting exponents during division and multiplication?

Tip:

Remember that when dividing powers with the same base, you subtract the exponents!