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Dec 10, 2025

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3 pages

Mastering Algebraic Expressions: A-Level Pure Maths Chapter 1

Ann

@ann_jznv

Algebraic expressions are the building blocks of advanced maths, and... Show more

1 / 3

Index laws
Pours of same base simplify:
• amxa^ =am+n
• am ÷ an = am-n
• (am)^
= amn
• (ab)^ = a^bn
Eg
1- Algebitur expressions
Q 95-5x3
3x

Index Laws and Expanding Brackets

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

When you're dividing algebraic fractions, split them up term by term. For example, $\frac{9x^5 - 5x^3}{3x}$ becomes $\frac{9x^5}{3x} - \frac{5x^3}{3x} = 3x^4 - \frac{5}{3}x^2$ .

Expanding brackets means multiplying every term in one bracket by every term in another. With multiple brackets like $(2x + 5y)(3x - y)(2x + y)$ , 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!

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 $15x^2 + 42x - 9$ to get $3(5x^2 + 14x - 3)$ .

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

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

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

Surds and Rationalising Denominators

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

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

Rationalising denominators gets rid of surds from the bottom of fractions. For simple cases like $\frac{1}{\sqrt{a}}$ , multiply top and bottom by $\sqrt{a}$ . For expressions like $\frac{1}{a + \sqrt{b}}$ , multiply by the conjugate $(a - \sqrt{b})$ - 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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268

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Dec 10, 2025

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3 pages

Mastering Algebraic Expressions: A-Level Pure Maths Chapter 1

Ann

@ann_jznv

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.

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Index Laws and Expanding Brackets

When you're dividing algebraic fractions, split them up term by term. For example, $\frac{9x^5 - 5x^3}{3x}$ becomes $\frac{9x^5}{3x} - \frac{5x^3}{3x} = 3x^4 - \frac{5}{3}x^2$ .

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

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Factorising and Index Rules

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

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Surds and Rationalising Denominators

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