Let's simplify each of the expressions shown in the image and express the answers using positive indices.

(a) $2^{\frac{1}{3}} \times 2^{\frac{2}{3}}$

• By the law of exponents, $a^m \times a^n = a^{m+n}$: $2^{\frac{1}{3}} \times 2^{\frac{2}{3}} = 2^{\frac{1}{3} + \frac{2}{3}} = 2^1 = 2$

(b) $\frac{3^{\frac{1}{4}}}{3^{\frac{1}{2}}}$

• By the law of exponents $\frac{a^m}{a^n} = a^{m-n}$: $\frac{3^{\frac{1}{4}}}{3^{\frac{1}{2}}} = 3^{\frac{1}{4} - \frac{1}{2}} = 3^{\frac{1}{4} - \frac{2}{4}} = 3^{-\frac{1}{4}} = \frac{1}{3^{\frac{1}{4}}}$

(c) $5^{-\frac{1}{6}} \times 5^{\frac{2}{3}}$

• Again, by the law $a^m \times a^n = a^{m+n}$: $5^{-\frac{1}{6}} \times 5^{\frac{2}{3}} = 5^{-\frac{1}{6} + \frac{2}{3}} = 5^{-\frac{1}{6} + \frac{4}{6}} = 5^{\frac{3}{6}} = 5^{\frac{1}{2}} = \sqrt{5}$

(d) $7^{-\frac{2}{3}} \times 7^{\frac{3}{2}} \times 7$

• Simplify the exponents: $7^{-\frac{2}{3}} \times 7^{\frac{3}{2}} \times 7 = 7^{-\frac{2}{3}} \times 7^{\frac{3}{2}} \times 7^1$
• Combine exponents: $7^{-\frac{2}{3} + \frac{3}{2} + 1} = 7^{-\frac{2}{3} + \frac{3}{2} + \frac{3}{3}} = 7^{\frac{9}{6}} = 7^{\frac{3}{2}}$

(e) $\frac{b^{\frac{2}{3}}}{b^{\frac{1}{6}}}$

• Apply the quotient rule $\frac{a^m}{a^n} = a^{m-n}$: $\frac{b^{\frac{2}{3}}}{b^{\frac{1}{6}}} = b^{\frac{2}{3} - \frac{1}{6}} = b^{\frac{4}{6} - \frac{1}{6}} = b^{\frac{3}{6}} = b^{\frac{1}{2}} = \sqrt{b}$

Would you like more details or have any questions on any of these steps? Let me know!

Here are some related questions:

1. What is the general rule for multiplying expressions with the same base?
2. How do you simplify expressions with fractional exponents?
3. What is the rule for dividing powers with the same base?
4. How do negative exponents affect the result of simplification?
5. How do we convert fractional exponents to radical form?

Tip: Always simplify fractional exponents by finding a common denominator to make adding or subtracting easier!