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Maths272Updated 25 Sept 20263 pages

Mastering Algebraic Expressions: A-Level Pure Maths Chapter 1

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Algebraic expressions are the building blocks of advanced maths, and mastering them now will make A-level topics so much easier. This guide covers the essential skills you'll use constantly: index laws, expanding brackets, factorising, and working with surds.
Chapter 1 - Algebraic expressions – page 1

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Index Laws and Expanding Brackets

Index laws are your best mates when simplifying expressions with powers. Remember these four golden rules: am×an=am+na^m \times a^n = a^{m+n}, am÷an=am−na^m \div a^n = a^{m-n}, (am)n=amn(a^m)^n = a^{mn}, and (ab)n=anbn(ab)^n = a^n b^n.

When you're dividing algebraic fractions, split them up term by term. For example, 9x5−5x33x\frac{9x^5 - 5x^3}{3x} becomes 9x53x−5x33x=3x4−53x2\frac{9x^5}{3x} - \frac{5x^3}{3x} = 3x^4 - \frac{5}{3}x^2.

Expanding brackets means multiplying every term in one bracket by every term in another. With multiple brackets like (2x+5y)(3x−y)(2x+y)(2x + 5y)(3x - y)(2x + y), work systematically - expand two brackets first, then multiply by the third. Keep track of like terms and combine them at the end.

Top Tip: When expanding complex expressions, write out each step clearly to avoid silly mistakes - your future self will thank you!

Chapter 1 - Algebraic expressions – page 2

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Factorising and Index Rules

Factorising is basically expanding backwards - you're looking for common factors and patterns. Always start by taking out the highest common factor, like pulling out the 3 from 15x2+42x−915x^2 + 42x - 9 to get 3(5x2+14x−3)3(5x^2 + 14x - 3).

For quadratics, you need two numbers that multiply to give the constant term and add to give the middle coefficient. In x2+9x+20x^2 + 9x + 20, you want numbers that multiply to 20 and add to 9 - that's 4 and 5, giving (x+4)(x+5)(x+4)(x+5).

Negative and fractional indices follow the same rules but with twists: a−m=1ama^{-m} = \frac{1}{a^m}, a0=1a^0 = 1, and amn=amna^{\frac{m}{n}} = \sqrt[n]{a^m}. When solving equations like 27x3=1x27\sqrt[3]{x} = \frac{1}{x}, rewrite everything using fractional indices first.

Remember: Factorising completely means you can't factor any further - always double-check your answer!

Chapter 1 - Algebraic expressions – page 3

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Surds and Rationalising Denominators

Surds are square roots that can't be simplified to whole numbers, and they follow specific manipulation rules: ab=a×b\sqrt{ab} = \sqrt{a} \times \sqrt{b} and ab=ab\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}.

To simplify surds like 75\sqrt{75}, look for perfect square factors. Since 75=25×375 = 25 \times 3, we get 75=53\sqrt{75} = 5\sqrt{3}. You can then combine like terms: 53−23=335\sqrt{3} - 2\sqrt{3} = 3\sqrt{3}.

Rationalising denominators gets rid of surds from the bottom of fractions. For simple cases like 1a\frac{1}{\sqrt{a}}, multiply top and bottom by a\sqrt{a}. For expressions like 1a+b\frac{1}{a + \sqrt{b}}, multiply by the conjugate (a−b)(a - \sqrt{b}) - this creates a difference of squares that eliminates the surd.

Pro Strategy: Always multiply by the conjugate when you see addition or subtraction with surds in the denominator!

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