Ch 2 · 2.3–2.4 Algebraic Fractions and Indices II · IGCSE Mathematics 0580
Topic 2 of 9 in Algebra and Graphs — IGCSE Mathematics 0580 — 58 lessons.
Introduction
Algebraic fractions follow exactly the same rules as number fractions: common denominators to add, cancel common factors to simplify. The index laws from Number now apply to letters, and they let you solve equations like 5^x+1 = 25^x without logarithms. (Syllabus sections 2.3 and 2.4.)
Key ideas — Adding and subtracting
Use a common denominator, then combine numerators, bracketing anything being subtracted:
🧑🏫 Teacher's explanation
Adding algebraic fractions is exactly like adding regular fractions like 1/2 + 1/3. You cannot simply add the tops and bottoms. You must find a 'common language' for the denominators so they speak the same scale. Once they match, you combine the numerators.
📖 Definitions
🔍 Worked example — Easy
Simplify x/4 + x+2/3
🔍 Worked example — Standard
Simplify 3/x-1 - 2/x+2
🔍 Worked example — Exam-style
Simplify x+1/x^2-9 - 1/x-3
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Multiplying and dividing
Multiply numerators and denominators; to divide, multiply by the reciprocal. Cancel common factors first: 3a/4 × 8/9a^2 = 2/3a; 3a/4 ÷ 9a/2 = 3a/4 × 2/9a = dfrac16.
🧑🏫 Teacher's explanation
Think of algebraic fractions like slices of a pizza. If you have 1/2 of a pizza and you want to multiply it by 1/3, you are essentially taking a third of a half, which results in 1/6. In algebra, we do the exact same thing, but instead of just numbers, we have variables like x or a attached to them.
📖 Definitions
🔍 Worked example — Easy
Simplify 5x^2/3y × 9y^3/10x
🔍 Worked example — Standard
Simplify 4a^2/b ÷ 8a/b^3
🔍 Worked example — Exam-style
Simplify x^2 - 9/x^2 + 5x + 6 ÷ x - 3/x + 2
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
2x^2 - 8/3x + 6 × x + 3/x - 2 [3 marks]
Key ideas — Simplifying rational expressions
Factorise top and bottom fully, then cancel common brackets:
🧑🏫 Teacher's explanation
Think of an algebraic fraction like a large LEGO structure built from smaller blocks. You cannot just pull out a single red brick from the middle of a wall; you can only remove an entire pre-assembled block that is identical in both the numerator and the denominator. In algebra, these 'blocks' are called factors.
📖 Definitions
🔍 Worked example — Easy
Simplify 10a^3/5a
🔍 Worked example — Standard
Simplify x^2 - 9/x + 3
🔍 Worked example — Exam-style
Simplify 2x^2 - 8/x^2 - 4x
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Indices in algebra
The rules from 1.7 apply to letters: 3x^-2 × 4x^5 = 12x^3; 10x^5 ÷ 2x^-2 = 5x^7; (2x^3)^2 = 4x^6 (square the 2 as well); (8x^6)^frac23 = 4x^4.
🧑🏫 Teacher's explanation
Think of indices (or exponents) as a shorthand code for repeated multiplication. When we deal with algebraic terms like x^3, we are simply saying 'x is multiplied by itself three times'. The rules of indices are just shortcuts to help us manage these long strings of multiplication without writing them all out.
📖 Definitions
🔍 Worked example — Easy
Simplify 5x^4 × 3x^-2
🔍 Worked example — Standard
Simplify 20a^6/5a^8
🔍 Worked example — Exam-style
Solve for x: 2^3x-1 = 16
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Exponential equations
Write both sides as powers of the same base, then equate the indices:
🧑🏫 Teacher's explanation
Imagine you are trying to find a secret code for a lock. The lock has two sides, and the only way to make them match is to make sure the 'base' numbers are identical. Once the bases match, the exponents (the little numbers on top) must also be identical for the equation to hold true. It is like saying if 2^a = 2^b, then a must equal b because there is only one way to get that specific result from that specific base.
📖 Definitions
🔍 Worked example — Easy
Solve 2^3x-1 = 8
🔍 Worked example — Standard
Solve 9^x+2 = 27^x
🔍 Worked example — Exam-style
Solve 5^2x-1 = 1/125
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked examples
2x/3 - x - 5/4 = 8x - 3(x - 5)/12 = 5x + 15/12 = 5(x + 3)/12.
🔍 Worked example — Easy
Simplify 2x^2 - 8/x^2 + x - 6
🔍 Worked example — Standard
Solve 9^x = 27^x - 1
🔍 Worked example — Exam-style
Simplify 3a^4 × 9a/3a^4 ÷ 9a
⚠️ Common errors
Common mistakes
Practice
Answers: (1) 5(x + 1) + 2(2x - 1)/10 = 9x + 3/10. (2) (x - 3)(x + 3)/(x + 2)(x + 3) = x - 3/x + 2. (3) 2x^5/y^2. (4) 2^2x = 2^3 ⇒ x = tfrac32. (5) 3x^3.