https://file.aiquickdraw.com/mathbot/file/172609019346379quydlt.jpg

Let's go through the problems in the image. Each expression needs to be simplified with positive exponents.

### 10. \( 5x^4(x^2) \)
We multiply the powers of \( x \) since they have the same base:
\[
5x^4(x^2) = 5x^{4+2} = 5x^6
\]

### 11. \( 6y^2(2y^4)^2 \)
We need to square \( 2y^4 \) and then multiply the result by \( 6y^2 \):
\[
(2y^4)^2 = 4y^8
\]
Now multiply by \( 6y^2 \):
\[
6y^2 \times 4y^8 = 24y^{2+8} = 24y^{10}
\]

### 12. \( 10(x^2)^2 \)
First, apply the power of a power rule:
\[
(x^2)^2 = x^{2 \times 2} = x^4
\]
Then multiply by 10:
\[
10x^4
\]

### 13. \( \frac{7x^2}{x^{-3}} \)
Subtract the exponents in the division:
\[
x^{2 - (-3)} = x^{2 + 3} = x^5
\]
So, the expression becomes:
\[
7x^5
\]

### 14. \( \frac{12(x + y)^3}{9(x + y)} \)
Simplify the expression by canceling out \( (x + y) \):
\[
\frac{12(x + y)^3}{9(x + y)} = \frac{12}{9}(x + y)^{3-1} = \frac{4}{3}(x + y)^2
\]

### 15. \( \frac{3x \sqrt{x}}{x^{1/2}} \)
We know that \( \sqrt{x} = x^{1/2} \), so the expression becomes:
\[
\frac{3x \cdot x^{1/2}}{x^{1/2}} = 3x
\]

### 16. \( \left( \frac{\sqrt{2} \sqrt{x^3}}{\sqrt{x}} \right)^4 \)
First, express everything in terms of exponents. Recall that \( \sqrt{a} = a^{1/2} \):
\[
\sqrt{2} = 2^{1/2}, \quad \sqrt{x^3} = x^{3/2}, \quad \sqrt{x} = x^{1/2}
\]
So the expression inside the parentheses becomes:
\[
\frac{2^{1/2} x^{3/2}}{x^{1/2}} = 2^{1/2} x^{(3/2 - 1/2)} = 2^{1/2} x^1 = 2^{1/2} x
\]
Now, raise everything to the power of 4:
\[
\left( 2^{1/2} x \right)^4 = 2^{(1/2) \times 4} x^4 = 2^2 x^4 = 4x^4
\]

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If you'd like any further explanations or details on any of these problems, feel free to ask!

Here are five related questions:
1. How do you apply the laws of exponents for products and quotients?
2. What is the significance of expressing exponents as positive?
3. How does the power of a product rule work in expressions?
4. What happens when you raise a fraction with variables to a power?
5. How can you simplify radical expressions involving variables?

**Tip:** When simplifying exponents, always remember to add them when multiplying terms with the same base and subtract them when dividing.

Simplify each expression and write with positive exponents, including problems such as 5x^4(x^2) and more.

This math problem set focuses on simplifying exponential expressions, including applying the laws of exponents to expressions like 5x^4(x^2) and others. Key concepts include multiplication and division of powers, the power of a power rule, and simplifying radical expressions. Suitable for students in Grades 8-10.

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