Rational and Irrational
● Some numbers are irrational – cannot be expressed as fractions as
decimals go on forever without pattern
● Square root of number that isn’t perfect square is also irrational
● However this is partly rational as you can write it as the square root of a
rational number
● These numbers are called surds
Simplifying Surds
● Take the number inside the square root function and write it in terms of its
prime factors
● Find any prime factors that are raised to a power greater than one
● Take those factors out of the square root
Example
Adding and Subtracting Surds
● Treat like algebra
● Simplify surds
● Collect like terms
● Like terms are any rational numbers or any surds involving the same
number
Multiplying Surds
● Multiply surds as you would multiply any numbers
● Simplify surds
● Multiply the numbers within the square roots
● Simplify final answer
Rationalising the Denominator
● Surds can sometimes appear in the denominator of a fraction → this is a
pain
● Rationalising denominator = removing surds from denominator
● If denominator only includes a single surd and nothing else (or a surd
multiplied by a rational number) multiply top and bottom by that surd
● If denominator is in the form a + b where a and/or b are surds, multiply top
and bottom by (a – b)
● Simplify
The Rules of Indices
1. am × an = am + n
2. am ÷ an = am – n
3. (am)n = a m/n
4. a–n = 1
an
5. a0 = 1
6. a 1/n = √n a
7. a m/n = (√n a)m = √n am