Let's solve problems 7, 8, and 9 from the image.

Problem 7

Expression: $\left( 4a^2b^3 \right)^{\frac{1}{2}}$

Solution: To simplify this, we apply the power rule $(x^m)^n = x^{m \cdot n}$.

1. Simplify each part separately: $4^{\frac{1}{2}} = \sqrt{4} = 2$ $(a^2)^{\frac{1}{2}} = a^{2 \cdot \frac{1}{2}} = a^1 = a$ $(b^3)^{\frac{1}{2}} = b^{3 \cdot \frac{1}{2}} = b^{1.5} = b^{3/2}$

2. Combine the results: $\left( 4a^2b^3 \right)^{\frac{1}{2}} = 2a b^{3/2}$

Problem 8

Expression: $\frac{12}{x^2}$

Solution: This expression is already in its simplest form, assuming no further information is provided. It represents a rational function and doesn't simplify further without additional context.

So, the answer is: $\frac{12}{x^2}$

Problem 9

Expression: $\sqrt{96}$

Solution: To simplify $\sqrt{96}$, factor 96 into its prime factors.