What is covered in Grade 10 Mathematics Exponents, Equations & Inequalities?

Laws of exponents (including rational exponents). Solve linear, quadratic and simultaneous equations. Solve linear inequalities with interval notation.

Is Grade 10 Exponents, Equations & Inequalities tested in Paper 1 or Paper 2?

Grade 10 Mathematics Exponents, Equations & Inequalities is assessed in Paper 1 of the end-of-year examinations, as per the CAPS Mathematics curriculum guidelines.

When is Exponents, Equations & Inequalities taught in Grade 10 Mathematics?

Exponents, Equations & Inequalities is taught in Term 1 of Grade 10 according to the CAPS Annual Teaching Plan (ATP) for Mathematics in South Africa.

Is Grade 10 Mathematics Exponents, Equations & Inequalities on the CAPS curriculum?

Yes. Exponents, Equations & Inequalities is part of the official CAPS Mathematics curriculum for Grade 10, Term 1 in South Africa. The content on MathSciBuddy follows the Annual Teaching Plan (ATP) set by the Department of Basic Education.

Is this Grade 10 Mathematics study material free?

Yes. All chapter notes, definitions, formulas, and worked examples on MathSciBuddy are completely free to read. No account or subscription is needed to access CAPS-aligned study notes.

Grade 10 Mathematics

Grade 10 · Term 1Mathematics

Exponents, Equations & Inequalities

We revise and extend the laws of exponents to rational exponents, then solve every type of equation met in Grade 10: linear, quadratic (by factorisation), simultaneous, and literal. We close by solving linear inequalities and displaying the solution on a number line.

2.1 Laws of Exponents 2.2 Equations 2.3 Linear Inequalities

Week 7

2.1 Laws of Exponents

Revise laws of exponents for integer exponents ($m,n\in\mathbb{Z}$)
Use the laws for rational exponents ($m,n\in\mathbb{Q}$)
Simplify and solve exponential expressions

🌍

Real-World Connection

Every memory card, hard drive, and phone plan is measured in powers of 2: 4 GB, 8 GB, 16 GB, 32 GB… When engineers combine two memory banks of $2^{10}$ bytes each, they know instantly: $2^{10}\times2^{10}=2^{20}$ bytes = 1 MB. The exponent law $a^m\cdot a^n=a^{m+n}$ is not an abstract rule — it is why all digital technology uses base-2 storage.

Product rule

a^m\cdot a^n = a^{m+n}

Same base; add exponents.

Quotient rule

\dfrac{a^m}{a^n} = a^{m-n}\;(a\neq0)

Same base; subtract exponents.

Power of a power

(a^m)^n = a^{mn}

Multiply exponents.

Power of a product

(ab)^n = a^n b^n

Distribute exponent over the product.

Negative exponent

a^{-n} = \dfrac{1}{a^n}\;(a\neq0)

Reciprocate; make exponent positive.

Zero exponent

a^0 = 1\;(a\neq0)

Any non-zero base to power 0 equals 1.

Rational exponent

a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m

Denominator = root index; numerator = power.

🚨 Common Mistake

$(-2)^4=16$ but $-2^4=-16$ . The base of an exponent is only what is in brackets. When in doubt, add brackets.

Worked Examples

Worked Example

Simplify using exponent laws

Problem

\dfrac{2^{n+2}\cdot4^{n-1}}{8^n}

Worked Example

Rational Exponents

Problem

\left(\dfrac{27}{8}\right)^{-2/3}

Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q

L2 · Routine Procedures2 Q

L3 · Complex Procedures2 Q

L4 · Problem Solving2 Q

L1 · Knowledge1 mark

Write

x^{-3}

without negative exponents.

L1 · Knowledge2 marks

Evaluate

\left(\dfrac{1}{4}\right)^{1/2}

L2 · Routine Procedures3 marks

Simplify

\dfrac{a^3\cdot a^{-1}}{a^{-2}}

L2 · Routine Procedures3 marks

Simplify

\dfrac{3^{x+1}-3^x}{3^x}

L3 · Complex Procedures4 marks

Simplify

\dfrac{2^{2x}-4}{2^x+2}

L3 · Complex Procedures5 marks

Simplify

\dfrac{6^n\cdot9}{2^{n-1}\cdot3^{n+2}}

L4 · Problem Solving5 marks

2^x=3

, find

4^x-2^{x+1}

L4 · Problem Solving6 marks

Solve for

x

2^{2x}-5\cdot2^x+4=0

Week 8

2.2 Equations

Revise and solve linear equations
Solve quadratic equations by factorisation
Solve simultaneous linear equations in two unknowns
Solve literal equations (changing the subject)
Solve word problems involving linear, quadratic or simultaneous equations

🌍

Real-World Connection

A pharmacist mixing a solution needs an exact concentration — too much or too little and the medicine is wrong. The equation $2x+15=63$ is that precise constraint: $x$ is the unknown volume and the equation says the total must be exactly 63 ml. Solving it finds the one measurement that satisfies the prescription — and any method is valid as long as you treat both sides of the scale equally.

Property / Rule

Zero Product Property

If a product equals zero, at least one factor must be zero. This is the foundation for solving quadratic equations by factorisation.

AB=0 \Rightarrow A=0 \text{ or } B=0

Property / Rule

Solving simultaneous linear equations

Substitution: express one variable in terms of the other from one equation, then substitute into the second. Elimination: make coefficients of one variable equal and subtract (or add) the equations.

\begin{cases} y=2x+1 \\ 3x-y=4 \end{cases}

Property / Rule

Literal equations (changing the subject)

Use the same algebraic operations as solving any equation, but the answer will be in terms of other letters rather than numbers.

A=\pi r^2 \Rightarrow r = \sqrt{\dfrac{A}{\pi}}

🚨 Common Mistake

ALWAYS rearrange a quadratic to standard form $ax^2+bx+c=0$ BEFORE factorising. Never divide both sides by $x$ — this loses the solution $x=0$ .

Worked Examples

Worked Example

Solve a quadratic by factorisation

Problem

x^2-x=6

Worked Example

Simultaneous linear equations

Problem

2x+y=7

Worked Example

Literal equation — changing the subject

Problem

h

Worked Example

Word problem — setting up and solving an equation

Problem

The sum of three consecutive integers is 57. Find the three integers.

Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q

L2 · Routine Procedures2 Q

L3 · Complex Procedures2 Q

L4 · Problem Solving2 Q

L1 · Knowledge2 marks

Solve

3x-7=2

L1 · Knowledge2 marks

Solve

x^2=16

L2 · Routine Procedures3 marks

Solve

x^2+5x-14=0

by factorisation.

L2 · Routine Procedures4 marks

Solve simultaneously:

y=3x-1

and

2x+y=9

L3 · Complex Procedures5 marks

Solve

\dfrac{2x}{x-1}-\dfrac{x}{x+1}=\dfrac{4}{x^2-1}

x\neq\pm1

L3 · Complex Procedures4 marks

Make

r

the subject of

V=\dfrac{4}{3}\pi r^3

L4 · Problem Solving5 marks

Twice a number squared is 5 more than 9 times the number. Find all possible values.

L4 · Problem Solving5 marks

For what value(s) of

k

does

kx^2-6x+9=0

have equal roots?

Week 9

2.3 Linear Inequalities

Solve linear inequalities and show the solution graphically on a number line
Work with compound inequalities (AND/OR)
Express solutions using interval notation

🌍

Real-World Connection

An inequality is not a single answer but a region of answers. 'You must be taller than 1·5 m to ride this attraction' is an inequality — it describes all heights above 1·5 m, not one specific height.

Property / Rule

Inequality Rules

You can add/subtract the same quantity on both sides. Multiplying or dividing by a POSITIVE number preserves the direction. Multiplying or dividing by a NEGATIVE number REVERSES the direction.

\text{If }a>b\text{ and }c<0,\text{ then }ac<bc

Property / Rule

Number line and interval notation

Open circle ○ at an endpoint means that value is EXCLUDED ( $<$ or $>$ ). Closed circle ● means INCLUDED ( $\leq$ or $\geq$ ). Interval notation: round bracket for excluded, square bracket for included.

x>3:\;x\in(3;\infty) \qquad x\leq-1:\;x\in(-\infty;-1]

🚨 Common Mistake

THE most common error: forgetting to flip the sign when multiplying or dividing by a negative number. $-2x<6$ does NOT give $x<-3$ ; it gives $x>-3$ .

Worked Examples

Worked Example

Compound inequality

Problem

-3 < 2x+1 \leq 7

Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q

L2 · Routine Procedures2 Q

L3 · Complex Procedures2 Q

L4 · Problem Solving2 Q

L1 · Knowledge2 marks

x=5

a solution to

3x-4>11

L1 · Knowledge1 mark

What happens to the inequality sign when you divide both sides by

-4

L2 · Routine Procedures3 marks

Solve

5x+2\geq3x-6

and represent on a number line.

L2 · Routine Procedures4 marks

Solve

-2\leq\dfrac{x-3}{2}<4

. Express in interval notation.

L3 · Complex Procedures5 marks

Solve

\dfrac{3x-1}{4}>\dfrac{2x+3}{3}

L3 · Complex Procedures4 marks

Find integer values of

x

satisfying

-5<3-2x\leq9

L4 · Problem Solving5 marks

A rectangle has length

2x+3

and width

x+1

. For what values of

x

is the perimeter greater than 38?

L4 · Problem Solving4 marks

For what real values of

x

\dfrac{1}{x+2}>0

Algebraic Expressions

Trigonometry