The image shows a series of mathematical expressions involving radicals and exponents. Let's simplify them step by step.

(a) $\sqrt{x^3} \times \sqrt[3]{x^2}$

We can express both square roots and cube roots as fractional exponents: $\sqrt{x^3} = x^{3/2}, \quad \sqrt[3]{x^2} = x^{2/3}$ Now multiply the two expressions: $x^{3/2} \times x^{2/3} = x^{3/2 + 2/3}$ To add the exponents, find a common denominator: $\frac{3}{2} = \frac{9}{6}, \quad \frac{2}{3} = \frac{4}{6}, \quad \text{so} \quad \frac{9}{6} + \frac{4}{6} = \frac{13}{6}$ Thus, the result is: $x^{13/6}$

(c) $\frac{\sqrt{x}}{x} \times \sqrt[3]{x}$

First, express the square and cube roots as fractional exponents: $\frac{\sqrt{x}}{x} = \frac{x^{1/2}}{x} = x^{1/2 - 1} = x^{-1/2}, \quad \sqrt[3]{x} = x^{1/3}$ Now multiply the two expressions: $x^{-1/2} \times x^{1/3} = x^{-1/2 + 1/3}$ To add the exponents, find a common denominator: $\frac{-1}{2} = \frac{-3}{6}, \quad \frac{1}{3} = \frac{2}{6}, \quad \text{so} \quad \frac{-3}{6} + \frac{2}{6} = \frac{-1}{6}$ Thus, the result is: $x^{-1/6}$

(e) $\frac{\sqrt{xy}}{y} \times \sqrt[3]{xy}$

Express the square root and cube root: $\frac{\sqrt{xy}}{y} = \frac{(xy)^{1/2}}{y} = x^{1/2} y^{1/2 - 1} = x^{1/2} y^{-1/2}, \quad \sqrt[3]{xy} = (xy)^{1/3} = x^{1/3} y^{1/3}$ Now multiply the two expressions: $x^{1/2} y^{-1/2} \times x^{1/3} y^{1/3} = x^{1/2 + 1/3} y^{-1/2 + 1/3}$ Add the exponents for both $x$ and $y$: $x: \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}, \quad y: \frac{-1}{2} + \frac{1}{3} = \frac{-3}{6} + \frac{2}{6} = \frac{-1}{6}$ Thus, the result is: $x^{5/6} y^{-1/6}$

(g) $\frac{y^2}{\sqrt{xy^3}} \times \frac{x}{\sqrt[3]{x^2y}}$

First, express all square and cube roots: $\frac{y^2}{\sqrt{xy^3}} = \frac{y^2}{(xy^3)^{1/2}} = \frac{y^2}{x^{1/2} y^{3/2}} = x^{-1/2} y^{2 - 3/2} = x^{-1/2} y^{1/2}$ $\frac{x}{\sqrt[3]{x^2y}} = \frac{x}{(x^2 y)^{1/3}} = \frac{x}{x^{2/3} y^{1/3}} = x^{1 - 2/3} y^{-1/3} = x^{1/3} y^{-1/3}$ Now multiply the two expressions: $x^{-1/2} y^{1/2} \times x^{1/3} y^{-1/3} = x^{-1/2 + 1/3} y^{1/2 - 1/3}$ Add the exponents for both $x$ and $y$: $x: \frac{-1}{2} + \frac{1}{3} = \frac{-3}{6} + \frac{2}{6} = \frac{-1}{6}, \quad y: \frac{1}{2} - \frac{1}{3} = \frac{3}{6} - \frac{2}{6} = \frac{1}{6}$ Thus, the result is: $x^{-1/6} y^{1/6}$

Would you like more details or have any questions? Here are some follow-up questions:

1. How do you convert a radical expression to a fractional exponent?
2. What are the rules for multiplying powers with the same base?
3. How can you simplify expressions with negative exponents?
4. What is the common denominator technique in adding fractions, and why is it useful here?
5. How do fractional exponents relate to radicals?

Tip: When simplifying radical expressions, always convert them to fractional exponents first; it often makes the operations easier to handle!