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• A-Level Maths

# Indices & Algebraic Methods

When working with indices, there are eight laws that must be followed:

These can be used to factorise and expand expressions.

### Surds

Surds are examples of irrational numbers, meaning they do not follow a repeating pattern but go on forever, uniquely. Pi is the most common example of an irrational number, but surds are slightly different - they are the square roots of non-square numbers.

√4 = 2 4 is a square number, so gives a rational square root
√2 = 1.4142... 2 is not a square number, so its square root is irrational

Like with indices, there are rules that apply to surds:

These can be used to rationalise denominators:

• For fractions in the form 1 / √a, multiply both numerator and denominator by √a

• For fractions in the form 1 / (a + √b), multiply both numerator and denominator by (a - √b)

• For fractions in the from 1 / (a - √b), multiply both numerator and denominator by (a + √b)

This is known as the conjugate pair (switching the sign of the denominator)

## Algebraic Fractions

To simplify algebraic fractions, factorise whatever can be factorised so that parts of the numerator and denominator can cancel:

### Multiplication

To multiply fractions, any common factors can be cancelled before multiplying the numerators and denominators.