Let's go through the first part of the exercise (7), where you're asked to simplify each expression and express the answer using positive indices.

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

• $\sqrt{x^3} = x^{3/2}$
• $\sqrt[3]{x^2} = x^{2/3}$ Now, multiply the terms: $x^{3/2} \times x^{2/3} = x^{\frac{3}{2} + \frac{2}{3}}$ Find a common denominator: $\frac{3}{2} + \frac{2}{3} = \frac{9}{6} + \frac{4}{6} = \frac{13}{6}$ Thus, the answer is $x^{13/6}$.

(b) $\sqrt[3]{xy} \times \sqrt{xy}$

• $\sqrt[3]{xy} = (xy)^{1/3} = x^{1/3}y^{1/3}$
• $\sqrt{xy} = (xy)^{1/2} = x^{1/2}y^{1/2}$ Now, multiply: $(x^{1/3}y^{1/3}) \times (x^{1/2}y^{1/2}) = x^{1/3 + 1/2}y^{1/3 + 1/2}$ Find common denominators: $1/3 + 1/2 = \frac{2}{6} + \frac{3}{6} = \frac{5}{6}$ Thus, the answer is $x^{5/6}y^{5/6}$, or $(xy)^{5/6}$.

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

• $\frac{\sqrt{x}}{x} = \frac{x^{1/2}}{x} = x^{1/2 - 1} = x^{-1/2}$
• $\sqrt[3]{x} = x^{1/3}$ Now, multiply: $x^{-1/2} \times x^{1/3} = x^{-1/2 + 1/3}$ Find a common denominator: $-1/2 + 1/3 = \frac{-3}{6} + \frac{2}{6} = \frac{-1}{6}$ Thus, the answer is $x^{-1/6}$.

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

• $\frac{x}{\sqrt[3]{x}} = \frac{x}{x^{1/3}} = x^{1 - 1/3} = x^{2/3}$
• $\sqrt{x} = x^{1/2}$ Now, multiply: $x^{2/3} \times x^{1/2} = x^{2/3 + 1/2}$ Find a common denominator: $2/3 + 1/2 = \frac{4}{6} + \frac{3}{6} = \frac{7}{6}$ Thus, the answer is $x^{7/6}$.

Would you like to go over the next ones (e)-(h) in detail? Let me know if you have any questions so far.

Here are some related practice questions for you to try:

1. Simplify $\sqrt[4]{x^8} \times \sqrt[3]{x^6}$.
2. Express $\frac{x^3}{\sqrt{x^5}}$ with positive indices.
3. Simplify $\frac{\sqrt[3]{x^2}}{\sqrt{x}}$.
4. Combine $x^{1/4} \times x^{1/3} \times x^{1/6}$ using positive indices.
5. Simplify $\frac{\sqrt{x} \times \sqrt[3]{x}}{\sqrt[4]{x^3}}$.

Tip: When simplifying indices, always look for opportunities to combine terms by adding or subtracting the powers of the same base.