MATHEMATICS
Complete Study Guide — Everything Explained Simply
Grade 11 CAPS — South Africa
, CHAPTER 1: Algebra — Working with Numbers and Letters
What is Algebra?
Algebra uses letters (like x and y) to represent unknown numbers. It helps us solve problems where we don't
know all the values yet.
Real Life Example: If apples cost R5 each, and you spent R35, how many apples did you buy? 5x =
35, so x = 7 apples. That's algebra!
Exponents (Powers)
An exponent tells you how many times to multiply a number by itself. Example: 2³ = 2 × 2 × 2 = 8
Laws of Exponents — The Rules You MUST Know
Law Rule Example
Multiplication (same base) aᵐ × aⁿ = aᵐ⁺ⁿ x³ × x⁴ = x⁷
Division (same base) aᵐ ÷ aⁿ = aᵐ⁻ⁿ x⁶ ÷ x² = x⁴
Power of a power (aᵐ)ⁿ = aᵐˣⁿ (x²)³ = x⁶
Zero exponent a⁰ = 1 (always!) 5⁰ = 1, x⁰ = 1
Negative exponent a⁻ⁿ = 1/aⁿ x⁻² = 1/x²
Fraction exponent a^(1/n) = ⁿ√a x^(1/2) = √x (square root)
Distribution (ab)ⁿ = aⁿ × bⁿ (2x)³ = 8x³
Worked Example — Simplify: (x³y²)² ÷ x⁴y
Step 1: Apply power to bracket: x⁶y⁴ ÷ x⁴y
Step 2: Divide (subtract exponents): x⁶⁻⁴ × y⁴⁻¹ = x²y³
Answer: x²y³
Surds (Square Roots and Cube Roots)
A surd is a root that cannot be simplified to a whole number. Example: √2 = 1.414... (it goes on forever)
Rule Formula Example
Multiply surds √a × √b = √(ab) √3 × √12 = √36 = 6
Divide surds √a ÷ √b = √(a/b) √50 ÷ √2 = √25 = 5
Complete Study Guide — Everything Explained Simply
Grade 11 CAPS — South Africa
, CHAPTER 1: Algebra — Working with Numbers and Letters
What is Algebra?
Algebra uses letters (like x and y) to represent unknown numbers. It helps us solve problems where we don't
know all the values yet.
Real Life Example: If apples cost R5 each, and you spent R35, how many apples did you buy? 5x =
35, so x = 7 apples. That's algebra!
Exponents (Powers)
An exponent tells you how many times to multiply a number by itself. Example: 2³ = 2 × 2 × 2 = 8
Laws of Exponents — The Rules You MUST Know
Law Rule Example
Multiplication (same base) aᵐ × aⁿ = aᵐ⁺ⁿ x³ × x⁴ = x⁷
Division (same base) aᵐ ÷ aⁿ = aᵐ⁻ⁿ x⁶ ÷ x² = x⁴
Power of a power (aᵐ)ⁿ = aᵐˣⁿ (x²)³ = x⁶
Zero exponent a⁰ = 1 (always!) 5⁰ = 1, x⁰ = 1
Negative exponent a⁻ⁿ = 1/aⁿ x⁻² = 1/x²
Fraction exponent a^(1/n) = ⁿ√a x^(1/2) = √x (square root)
Distribution (ab)ⁿ = aⁿ × bⁿ (2x)³ = 8x³
Worked Example — Simplify: (x³y²)² ÷ x⁴y
Step 1: Apply power to bracket: x⁶y⁴ ÷ x⁴y
Step 2: Divide (subtract exponents): x⁶⁻⁴ × y⁴⁻¹ = x²y³
Answer: x²y³
Surds (Square Roots and Cube Roots)
A surd is a root that cannot be simplified to a whole number. Example: √2 = 1.414... (it goes on forever)
Rule Formula Example
Multiply surds √a × √b = √(ab) √3 × √12 = √36 = 6
Divide surds √a ÷ √b = √(a/b) √50 ÷ √2 = √25 = 5
