Let's go through the questions one by one and solve them.

1. Evaluate the following:

i. $(2^4)$

$2^4 = 16$

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

$3^2 = 9 \quad \text{so} \quad (3^2)^{-1} = \frac{1}{9}$

iii. $(2^3)^2 + 2^0$

$(2^3)^2 = 2^{6} = 64 \quad \text{and} \quad 2^0 = 1 \quad \text{so} \quad 64 + 1 = 65$

iv. $(5^2)^1 + \frac{1}{5}$

$(5^2)^1 = 25 \quad \text{and} \quad \frac{1}{5} = 0.2 \quad \text{so} \quad 25 + 0.2 = 25.2$

v. $(4^0) \times 1$

$4^0 = 1 \quad \text{so} \quad 1 \times 1 = 1$

vi. $(10^2)$

$10^2 = 100$

2. Simplify the expression using positive indices:

i. $(x^3)^4$

$(x^3)^4 = x^{12}$

ii. $(p^{-2})^2$

$(p^{-2})^2 = p^{-4}$

iii. $(a^2 b^2)^2$

$(a^2 b^2)^2 = a^4 b^4$

iv. $(2x^2)^3$

$(2x^2)^3 = 2^3 \cdot (x^2)^3 = 8x^6$

v. $\frac{x^3}{x}$

$\frac{x^3}{x} = x^{3-1} = x^2$

vi. $\left( \frac{a^3}{b^2} \right)^2$

$\left( \frac{a^3}{b^2} \right)^2 = \frac{a^6}{b^4}$

vii. $\left( \frac{m^3}{n^2} \right)^{-2}$

$\left( \frac{m^3}{n^2} \right)^{-2} = \frac{n^4}{m^6}$

viii. $(p^{-2})^{-4}$

$(p^{-2})^{-4} = p^{8}$

Would you like further clarification on any of the steps? Here are 5 related questions to practice further:

1. What is the value of $(5^3)^2 + 3^0$?
2. Simplify $\left( \frac{x^2}{y^3} \right)^3$.
3. Evaluate $(7^2)^0 + (3^1)^3$.
4. Simplify $\frac{a^4 b^3}{a^2 b^2}$.
5. Solve $4^{-2} + 2^3$.

Tip: When simplifying indices, remember to multiply the exponents when raising a power to another power!