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: $\left( 2^{1/2} x \right)^4 = 2^{(1/2) \times 4} x^4 = 2^2 x^4 = 4x^4$

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.