To find equivalent expressions for the area of a rectangle where the sides are $7^2$ and $4^2$, we follow these steps:

1. The formula for the area $A$ of a rectangle is: $A = \text{length} \times \text{width}$

2. In this case, the length and width are $7^2$ and $4^2$, so: $A = 7^2 \times 4^2$

3. Using the property of exponents $a^m \times b^m = (a \times b)^m$, we can simplify: $A = (7 \times 4)^2$

4. Simplifying further: $A = 28^2$

5. Finally, evaluating $28^2$: $A = 784$

So, the area of the rectangle can be expressed in three equivalent ways:

• $7^2 \times 4^2$
• $(7 \times 4)^2$
• $784$

Would you like a more detailed explanation or have any questions?

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Here are 5 related questions to expand your understanding:

1. What is the area of a rectangle with sides $6^2$ and $3^2$?
2. How does the property of exponents help simplify multiplication in geometry problems?
3. If one side of a rectangle is doubled, how does it affect the area?
4. Can you use the distributive property to break down $(7^2 + 4^2) \times 2$?
5. How do you calculate the area of a rectangle if one side is an expression, such as $x + 3$?

Tip: Remember that knowing exponent rules can simplify calculations involving geometric shapes.