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 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)
To simplify algebraic fractions, factorise whatever can be factorised so that parts of the numerator and denominator can cancel:
To multiply fractions, any common factors can be cancelled before multiplying the numerators and denominators.