https://file.aiquickdraw.com/mathbot/file/1726575079606mp0pn22s.jpg

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!

1. Evaluate the following expressions and simplify using positive indices for the second set of problems.

This page explains how to evaluate powers and exponents, including simplifying expressions with indices. Learn how to apply the laws of exponents with examples such as (2^3)^2, (p^-2)^4, and more. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation