Exponents and Surds

Master the laws of exponents and operations with surds

CAPS Grade 10 Mathematics

This topic forms part of the CAPS-aligned Grade 10 Mathematics curriculum and provides fundamental skills for algebra and higher mathematics. Each section includes interactive games and quizzes to test your understanding.

Learning Outcomes

  • Understand and apply the laws of exponents
  • Simplify expressions with integer exponents
  • Solve basic exponential equations
  • Identify and simplify surds
  • Perform operations with surds
  • Rationalize denominators containing surds

Introduction to Exponents

An exponent indicates how many times a base number is multiplied by itself.

Exponent Notation
aⁿ = a × a × a × ... × a (n times)

Basic Example

Example

Calculate 2³

Solution
2³ = 2 × 2 × 2 = 8

Base = 2, Exponent = 3, Result = 8

Quiz 1 - Basic Exponents

What is 3⁴?

A) 12
B) 27
C) 81
D) 64

Laws of Exponents

1

Product of Powers

aᵐ × aⁿ = aᵐ⁺ⁿ

x² × x³ = ?

x⁵
2

Quotient of Powers

aᵐ ÷ aⁿ = aᵐ⁻ⁿ

y⁵ ÷ y² = ?

y³
3

Power of a Power

(aᵐ)ⁿ = aᵐⁿ

(z³)² = ?

z⁶
4

Power of a Product

(ab)ⁿ = aⁿbⁿ

(2a)³ = ?

8a³
5

Power of a Quotient

(a/b)ⁿ = aⁿ/bⁿ

(x/y)² = ?

x²/y²
6

Zero & Negative

a⁰ = 1 (a≠0)
a⁻ⁿ = 1/aⁿ

5⁰ = ?
3⁻² = ?

5⁰ = 1
3⁻² = 1/9

Quiz 2 - Laws of Exponents

Simplify: (x³)⁴

A) x⁷
B) x¹²
C) x⁸¹
D) x¹

Solving Exponential Equations

Example 1

Solve: 2ˣ⁺¹ = 8

  • 2ˣ⁺¹ = 2³
  • x + 1 = 3
  • x = 2
Example 2

Solve: 3²ˣ⁻¹ = 27

  • 3²ˣ⁻¹ = 3³
  • 2x - 1 = 3
  • 2x = 4 → x = 2

Quiz 3 - Exponential Equations

Solve for x: 5ˣ = 125

A) 2
B) 3
C) 4
D) 5

Introduction to Surds

Definition: A surd is an irrational number expressed as the root of an integer that cannot be simplified to a rational number.

Surds vs Non-Surds

  • Surds: √2, √3, √5, ∛7
  • Not Surds: √4 = 2, √9 = 3

Simplifying Surds

Simplify √12

√(4×3) = √4 × √3 = 2√3

Another Example

Simplify √50

√(25×2) = √25 × √2 = 5√2

Quiz 4 - Surds

Simplify √18

A) 3√2
B) 2√3
C) 9√2
D) √9

Operations with Surds

+

Addition & Subtraction

Only combine like surds

2√3 + 5√3

= 7√3
×

Multiplication

Multiply coefficients and surds separately

(2√3) × (3√5)

= 6√15
÷

Division

(6√10) ÷ (2√2)

= 3√5

Rationalizing the Denominator

Simple Denominator

Problem

Rationalize 1/√2

(1/√2) × (√2/√2) = √2/2

Binomial Denominator

Problem

Rationalize 2/(1 + √3)

= [2/(1+√3)] × [(1-√3)/(1-√3)]
= (2 - 2√3)/(1 - 3) = (2 - 2√3)/(-2) = -1 + √3

Quiz 5 - Rationalizing

Rationalize: 3/√3

A) √3
B) 3√3
C) √9
D) 1/√3

Practice & Assess

Match - Law to Name

aᵐ × aⁿ
Product of Powers
aᵐ ÷ aⁿ
Quotient of Powers
(aᵐ)ⁿ
Power of a Power
a⁰
Zero Exponent

Fill - Surd Simplification

√75 = ___√3

Summary of Key Concepts

Exponent Laws:
  • aᵐ × aⁿ = aᵐ⁺ⁿ
  • aᵐ ÷ aⁿ = aᵐ⁻ⁿ
  • (aᵐ)ⁿ = aᵐⁿ
  • a⁰ = 1, a⁻ⁿ = 1/aⁿ
Surd Rules:
  • √(ab) = √a × √b
  • Only like surds can be added/subtracted
  • Multiply coefficients and surd parts separately
  • Rationalize denominators to eliminate surds

Key Terms

Exponent Base Power Surd Rationalize Conjugate Product Rule Quotient Rule Zero Exponent Negative Exponent

Practice Questions

Q1

Simplify: (3x²y³)² × (2x⁴y)

Q2

Solve: 5ˣ⁻² = 125

Q3

Simplify: √72

Answers

Q1: 9x⁸y⁷ | Q2: x = 5 | Q3: 6√2

Algebraic Expressions Linear Inequalities