https://file.aiquickdraw.com/mathbot/file/1727180860082qkt7awbk.jpg

Let's work through each of the three problems step by step.

### 5) \((7y^4)^2\)

Using the property of exponents \((a^m)^n = a^{m \cdot n}\), we apply the exponent to both the number \(7\) and the variable \(y^4\):

\[
(7y^4)^2 = 7^2 \cdot (y^4)^2 = 49 \cdot y^{4 \cdot 2} = 49y^8
\]

So the simplified result is:
\[
49y^8
\]

---

### 7) \((2x^4 y^4)^3\)

Again, using the exponent rule \((a^m)^n = a^{m \cdot n}\), we apply the exponent to each factor inside the parentheses:

\[
(2x^4 y^4)^3 = 2^3 \cdot (x^4)^3 \cdot (y^4)^3 = 8 \cdot x^{4 \cdot 3} \cdot y^{4 \cdot 3} = 8x^{12}y^{12}
\]

So the simplified result is:
\[
8x^{12}y^{12}
\]

---

### 9) \((x^2 y^2)^2\)

Using the same exponent rule, we apply the outer exponent \(2\) to both the \(x^2\) and the \(y^2\):

\[
(x^2 y^2)^2 = (x^2)^2 \cdot (y^2)^2 = x^{2 \cdot 2} \cdot y^{2 \cdot 2} = x^4 y^4
\]

So the simplified result is:
\[
x^4 y^4
\]

---

Would you like a more detailed explanation for any of these steps?

Here are some related questions you might find interesting:
1. How would you simplify \((3a^2b^5)^3\)?
2. What is the result of \((5p^3q^2)^4\)?
3. Can you apply the same method to simplify \((4m^5n^3)^2\)?
4. What happens if one of the bases inside the parentheses is a fraction, e.g., \((\frac{1}{2}x^2)^3\)?
5. How would the results change if we had negative exponents, like \((x^{-3}y^{-2})^2\)?

**Tip:** Always remember that when raising a product to a power, you can distribute the exponent to each factor inside the parentheses!

Learn how to simplify expressions like (7y^4)^2, (2x^4 y^4)^3, and (x^2 y^2)^2 using the laws of exponents. This guide is suitable for students in Grades 8-10 and covers key algebraic concepts.

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