Ch 2 · 1.3, 1.7, 1.18 Powers, Roots, Indices and Surds · O Level Mathematics 4024

Topic 2 of 6 in Number — O Level Mathematics 4024 — 48 lessons.

Introduction

A square of side 13 cm has area 13^2 = 169 cm²; going back, √(169) = 13. Powers and roots undo each other, and index notation makes huge or tiny numbers manageable. Surds such as sqrt2 keep answers exact instead of rounding them. (Syllabus sections 1.3, 1.7 and 1.18.)

Key ideas — Powers and roots to know

Squares from 1^2 = 1 to 15^2 = 225 (e.g. 13^2 = 169, 14^2 = 196) and their square roots; cubes 1, 8, 27, 64, 125 and 10^3 = 1000 and their cube roots. Every positive number has two square roots (±), but √(\ ) means the positive one.

🧑‍🏫 Teacher's explanation

Think of indices (powers) as a shorthand for repeated multiplication. If a^3 means a × a × a, then the index tells you how many times the base is used in the multiplication. It is like a growth instruction: 2^2 is a square area, while 2^3 is a cubic volume.

📖 Definitions

🔍 Worked example — Easy

Simplify x^5 × x^-2 and find the value of 27^2/3.

🔍 Worked example — Standard

Simplify √(72) and express in the form a√(b) where b is a prime number.

🔍 Worked example — Exam-style

Rationalise the denominator of 5/3 + √(2)

⚠️ Common errors

🧠 Quick check 1

🧠 Quick check 2

🎯 Exam question

Key ideas — Rules of indices

For the same base a:

🧑‍🏫 Teacher's explanation

Think of indices (or exponents) as a shorthand way of writing repeated multiplication. Instead of writing 2 × 2 × 2 × 2, we write 2^4. The 'base' is the number being multiplied, and the 'index' tells you how many times to multiply it by itself.

📖 Definitions

🔍 Worked example — Easy

Simplify 5^3 × 5^-5

🔍 Worked example — Standard

Evaluate 64^2/3

🔍 Worked example — Exam-style

Simplify frac(2^3)^42^10

⚠️ Common errors

🧠 Quick check 1

🧠 Quick check 2

🎯 Exam question

Key ideas — Simplifying surds

A surd is a root that is irrational, like sqrt5. Use √(ab) = sqrt asqrt b and take out the largest square factor:

🧑‍🏫 Teacher's explanation

Think of a surd as a number that is 'stuck' inside a root sign because it isn't a perfect square. For example, √(25) is just 5, but √(20) is an irrational number that we write as a surd. Simplifying a surd is like 'unpacking' a suitcase; we want to find the largest square number hidden inside the root so we can pull it out.

📖 Definitions

🔍 Worked example — Easy

Simplify √(72)

🔍 Worked example — Standard

Simplify √(50) + √(18)

🔍 Worked example — Exam-style

Expand and simplify (3 - √(5))(3 + √(5))

⚠️ Common errors

🧠 Quick check 1

🧠 Quick check 2

🎯 Exam question

Key ideas — Rationalising the denominator

Remove a surd from the bottom of a fraction:

🧑‍🏫 Teacher's explanation

In mathematics, we prefer our fractions to look 'tidy'. Having a square root in the denominator (the bottom part) is like having a messy decimal that never ends; it makes adding or comparing fractions very difficult. Rationalising the denominator is the process of 'cleaning up' the fraction so that the denominator becomes a rational number (a whole number or a simple fraction).

Ch 2 · 1.3, 1.7, 1.18 Powers, Roots, Indices and Surds · O Level Mathematics 4024

All chapters in Number