Let's go through the simplification step-by-step for each part of the problem shown in the image. Here's the breakdown:

a) $\frac{6tw}{2t^{-3}w}$

• Cancel $w$ from the numerator and denominator: $\frac{6tw}{2t^{-3}w} = \frac{6t}{2t^{-3}}$
• Simplify the constants: $\frac{6}{2} = 3$
• Use the laws of exponents: $t^1 \div t^{-3} = t^{1 - (-3)} = t^4$: $3t^4$ Thus, the simplified expression is: $3t^4$

b) $\frac{4m \times \frac{1}{2} \times 3m^0}{6m \times \frac{1}{2}}$

• Remember that $m^0 = 1$, so the numerator simplifies to: $4m \times \frac{1}{2} \times 3 = 6m$
• Simplify the denominator: $6m \times \frac{1}{2} = 3m$
• Now, simplify the expression: $\frac{6m}{3m} = 2$ Thus, the simplified expression is: $2$

c) $\frac{8a \times \frac{1}{2}}{2a \times \frac{2}{3}}$

• Simplify the numerator: $8a \times \frac{1}{2} = 4a$
• Simplify the denominator: $2a \times \frac{2}{3} = \frac{4a}{3}$
• Now, divide: $\frac{4a}{\frac{4a}{3}} = 3$ Thus, the simplified expression is: $3$

d) $\frac{3k^2 \times 2k^{-5}}{6k^{-4}}$

• Multiply the terms in the numerator: $3k^2 \times 2k^{-5} = 6k^{-3}$
• Now simplify the fraction: $\frac{6k^{-3}}{6k^{-4}} = k^{-3 - (-4)} = k^1 = k$ Thus, the simplified expression is: $k$

e) $\frac{6t^{-2} + 8t}{2t^{-3}}$

• Factor the numerator: $\frac{6t^{-2} + 8t}{2t^{-3}} = \frac{2(3t^{-2} + 4t)}{2t^{-3}}$ Cancel out the 2: $\frac{3t^{-2} + 4t}{t^{-3}}$
• Now apply the exponent rule for division to each term: $\frac{3t^{-2}}{t^{-3}} = 3t^{1}, \quad \frac{4t}{t^{-3}} = 4t^4$
• Combine the terms: $3t + 4t^4$ Thus, the simplified expression is: $4t^4 + 3t$

Would you like me to continue with parts f, g, and h? Let me know if you want further explanations or have any questions!

Here are 5 questions to expand your understanding:

1. What are the laws of exponents used for simplifying expressions like in part a?
2. How do we deal with terms involving zero exponents, as in part b?
3. What methods help simplify complex fractions like in part c?
4. Why is it important to factor before simplifying, as done in part e?
5. Can you apply the same rules to variables raised to negative powers?

Tip: Always simplify the constants and variables separately before combining them into a final answer.