Grade 8-9 skills: proof, surds, algebraic fractions and functions — Edexcel GCSE Maths revision

Surds

• sqrt(a) × sqrt(b) = sqrt(ab); simplify by finding square factors, e.g. sqrt(50) = sqrt(25×2) = 5sqrt(2).
• Rationalise a denominator by multiplying top and bottom by the surd (or conjugate for a+sqrt(b) denominators).
• (a + sqrt(b))(a - sqrt(b)) = a^2 - b, useful for rationalising and proofs.

Algebraic fractions

• Factorise numerator and denominator fully before cancelling common factors.
• To add/subtract algebraic fractions, find a common denominator, exactly like numeric fractions.
• To solve an equation with algebraic fractions, multiply every term by the common denominator to clear fractions.

Key terms

Surd

An irrational root, e.g. sqrt(2), that cannot be simplified to a whole number.

Rationalise

To remove a surd from the denominator of a fraction.

Conjugate

For a+sqrt(b), the conjugate is a-sqrt(b); used to rationalise denominators.

Composite function

A function formed by applying one function then another, e.g. fg(x).

Inverse function

A function that reverses the effect of the original function, written f^-1(x).

Algebraic fraction

A fraction with algebraic expressions in the numerator and/or denominator.

Common exam traps

• Forgetting to fully simplify a surd (leaving a square factor inside the root).
• Rationalising with just the surd when the denominator is a+sqrt(b) (need the conjugate instead).
• Cancelling terms in algebraic fractions without factorising first.

Practice questions with answers

1. 1. Simplify sqrt(72)

   Answer: 6sqrt(2)

   72 = 36×2. sqrt(36×2) = 6sqrt(2).

2. 2. Simplify sqrt(3) × sqrt(12)

   Answer: 6

   sqrt(3×12)=sqrt(36)=6.

3. 3. Rationalise the denominator: 5/sqrt(5)

   Answer: sqrt(5)

   Multiply top and bottom by sqrt(5): 5sqrt(5)/5 = sqrt(5).

4. 4. Rationalise the denominator: 3/(2+sqrt(3))

   Answer: 6 - 3sqrt(3)

   Multiply by conjugate (2-sqrt(3)): 3(2-sqrt(3))/((2)^2-3) = (6-3sqrt(3))/1 = 6-3sqrt(3).

5. 5. Simplify (3 + sqrt(2))(3 - sqrt(2))

   Answer: 7

   Difference of squares: 3^2 - 2 = 9-2=7.

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