The problem consists of two questions. Question 1 asks to rewrite expressions with negative exponents as equivalent expressions with positive exponents. Question 2 asks to evaluate some expressions.

1. Problem Description

The problem consists of two questions.

Question 1 asks to rewrite expressions with negative exponents as equivalent expressions with positive exponents.

Question 2 asks to evaluate some expressions.

2. Solution Steps

Question 1:

i. $5^{-3} = \frac{1}{5^3}$

ii. $(\frac{1}{3})^{-2} = (\frac{3}{1})^2 = 3^2$

iii. $(0.2)^{-4} = (\frac{2}{10})^{-4} = (\frac{1}{5})^{-4} = (\frac{5}{1})^4 = 5^4$

iv. $x^{-4} = \frac{1}{x^4}$

v. $(\frac{2}{4})^{-2} = (\frac{1}{2})^{-2} = (\frac{2}{1})^2 = 2^2$

Question 2:

i. $4^3 \times 4 = 4^3 \times 4^1 = 4^{3+1} = 4^4 = 256$

ii. $3^5 \times 3^{-4} = 3^{5+(-4)} = 3^{5-4} = 3^1 = 3$

iii. $\frac{5^8}{5^8} = 5^{8-8} = 5^0 = 1$

3. Final Answer

Question 1:

i. $\frac{1}{5^3}$

ii. $3^2$

iii. $5^4$

iv. $\frac{1}{x^4}$

v. $2^2$

Question 2:

i. $256$

ii. $3$

iii. $1$