Let's solve the problem step by step, simplifying each expression using positive indices. The goal is to express everything with positive exponents.

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

• Convert to exponential form:

  • $\sqrt{x^3} = x^{3/2}$
  • $\sqrt[3]{x^2} = x^{2/3}$

• Multiply the terms: $x^{3/2} \times x^{2/3} = x^{(3/2 + 2/3)}$ Find a common denominator for $\frac{3}{2}$ and $\frac{2}{3}$, which is 6: $\frac{3}{2} = \frac{9}{6}, \quad \frac{2}{3} = \frac{4}{6}$ Now, add: $x^{\frac{9}{6} + \frac{4}{6}} = x^{\frac{13}{6}}$

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

• Convert to exponential form:

  • $\sqrt[3]{xy} = (xy)^{1/3}$
  • $\sqrt{xy} = (xy)^{1/2}$

• Multiply the terms: $(xy)^{1/3} \times (xy)^{1/2} = (xy)^{(1/3 + 1/2)}$ Find a common denominator for $\frac{1}{3}$ and $\frac{1}{2}$, which is 6: $\frac{1}{3} = \frac{2}{6}, \quad \frac{1}{2} = \frac{3}{6}$ Now, add: $(xy)^{\frac{2}{6} + \frac{3}{6}} = (xy)^{\frac{5}{6}}$

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

• Convert to exponential form:

  • $\sqrt{x} = x^{1/2}$
  • $\sqrt[3]{x} = x^{1/3}$
  • $\frac{\sqrt{x}}{x} = \frac{x^{1/2}}{x} = x^{1/2 - 1} = x^{-1/2}$

• Multiply the terms: $x^{-1/2} \times x^{1/3} = x^{(-1/2 + 1/3)}$ Find a common denominator for $-\frac{1}{2}$ and $\frac{1}{3}$, which is 6: $-\frac{1}{2} = -\frac{3}{6}, \quad \frac{1}{3} = \frac{2}{6}$ Now, add: $x^{-\frac{3}{6} + \frac{2}{6}} = x^{-\frac{1}{6}}$

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

• Convert to exponential form:

  • $\frac{x}{\sqrt[3]{x}} = \frac{x^1}{x^{1/3}} = x^{1 - 1/3} = x^{2/3}$
  • $\sqrt{x} = x^{1/2}$

• Multiply the terms: $x^{2/3} \times x^{1/2} = x^{(2/3 + 1/2)}$ Find a common denominator for $\frac{2}{3}$ and $\frac{1}{2}$, which is 6: $\frac{2}{3} = \frac{4}{6}, \quad \frac{1}{2} = \frac{3}{6}$ Now, add: $x^{\frac{4}{6} + \frac{3}{6}} = x^{\frac{7}{6}}$

Let me know if you'd like more details on the remaining parts (e)-(h), or if you have any questions so far. Here are some follow-up questions:

1. How do we determine the common denominator when adding fractions?
2. What is the difference between a square root and a cube root in terms of exponents?
3. How does simplifying fractional exponents help in simplifying expressions?
4. Why do we express square roots as powers of $\frac{1}{2}$?
5. How can we verify the final expressions using exponent rules?

Tip: Always convert roots to fractional exponents first when simplifying expressions involving radicals. This makes applying exponent rules much easier.