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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.

Simplify each of the following, expressing the answer using positive indices.

Learn how to simplify algebraic expressions with radicals and exponents using positive indices. This problem involves multiplying terms with roots and fractional exponents, suitable for students in Grades 9-12.

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