https://file.aiquickdraw.com/mathbot/file/1727806577150b0dgx13f.jpg

The equation provided in the image is:

\[
\frac{(2xy)^4}{4xy} = 4x^a y^b
\]

Let's simplify and solve for \( a \) and \( b \).

### Step 1: Simplify the left-hand side
First, expand \( (2xy)^4 \):
\[
(2xy)^4 = 2^4 \cdot x^4 \cdot y^4 = 16x^4y^4
\]

Now, divide this expression by \( 4xy \):
\[
\frac{16x^4y^4}{4xy} = \frac{16}{4} \cdot \frac{x^4}{x^1} \cdot \frac{y^4}{y^1} = 4x^{4-1}y^{4-1} = 4x^3y^3
\]

### Step 2: Compare both sides
Now that we have simplified the left-hand side:
\[
4x^3y^3 = 4x^a y^b
\]

Since the coefficients and bases are the same on both sides, we can equate the exponents:
\[
a = 3 \quad \text{and} \quad b = 3
\]

### Final Answer:
\[
a = 3, \quad b = 3
\]

Would you like more details or have any other questions?

Here are five related questions to expand on this:
1. How would the equation change if the exponent on \( (2xy) \) was different?
2. Can you generalize the solution for \( \frac{(2xy)^n}{4xy} = 4x^a y^b \)?
3. What happens if the denominator was \( 4x^2y^2 \) instead of \( 4xy \)?
4. How do exponent rules affect the simplification process in this type of equation?
5. What is the general form of dividing two expressions with exponents like \( \frac{x^m}{x^n} \)?

**Tip:** When simplifying expressions with exponents, always apply the laws of exponents carefully to combine terms efficiently.

Simplify the expression (2xy)^4 / 4xy = 4x^a y^b and solve for a and b.

Learn how to simplify the expression (2xy)^4 / 4xy and solve for the exponents a and b. This algebraic problem involves the laws of exponents and exponentiation rules. Suitable for students in Grades 9-12.

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