Simplifying Surds

Surds & Exact Values

What is a surd?

A surd is the square root of a non-square integer
Using surds lets you leave answers in exact form
- e.g. $5 square root of 2$ rather than $7.071067812$

Surd and not surd, A Level & AS Level Pure Maths Revision Notes

How do I calculate with surds?

Multiplying surds
- You can multiply numbers under square roots together
- eg. $square root of 3 space cross times space square root of 5 space equals square root of space 3 cross times 5 space end root equals space square root of 15$
Dividing surds
- You can divide numbers under square roots
- eg. $square root of 21 space divided by space square root of 7 equals space square root of 21 space divided by space 7 space end root equals space square root of 3$
Factorising surds
- You can factorise numbers under square roots
- eg. $square root of 35 space equals square root of 5 space cross times space 7 space end root equals space square root of 5 space cross times square root of 7$
Adding or subtracting surds is very like adding or subtracting letters in algebra – you can only add or subtract multiples of “like” surds
- eg. $3 square root of 5 plus space 8 square root of 5 space equals space 11 square root of 5 space o r space 7 square root of 3 space – space 4 square root of 3 space equals space 3 square root of 3$
- Be very careful here, you can not add or subtract numbers under square roots
- Think about $square root of 9 space end root plus space square root of 4 equals space 3 space plus space 2 space equals space 5$
  - It is not equal to $square root of 9 plus 4 end root space equals space square root of 13 space equals space 3.60555 horizontal ellipsis$

Exam Tip

If you are working on an exam question and your calculator gives you an answer as a surd, leave the value as a surd throughout the rest of your calculations to make sure you do not lose accuracy throughout your questions

Simplifying Surds

How do I simplify surds?

To simplify a surd, separate out a square factor and square root it
- Look for the greatest square number that is a factor of the number you are simplifying
- eg. $square root of 48 space equals space square root of 16 space cross times space 3 end root space equals space square root of 16 space cross times space square root of 3 equals space 4 space cross times space square root of 3 space equals space 4 square root of 3$

2-1-2-surds-simplify

You can collect like terms with surds like you do with letters in algebra
Understanding how to simplify surds can help reduce expressions and collect like terms
- e.g. simplify $square root of 32 space plus space square root of 8$ by simplifying each part separately
  $table row cell square root of 32 space plus space square root of 8 space end cell equals cell space square root of 16 space cross times space square root of 2 space plus space square root of 4 space cross times space square root of 2 space end cell row blank equals cell space 4 square root of 2 space plus space 2 square root of 2 space end cell row blank equals cell space 6 square root of 2 end cell end table$
An important skill is multiplying double brackets containing surds
- This can be done in the same way as multiplying out double brackets algebraically and simplifying
- The property $open parentheses square root of a close parentheses squared space equals space a$ can be used to simplify the expression, once expanded

Exam Tip

Worked example

Write $square root of 54 space minus space square root of 24$ in the form $p square root of q$ where $p$ Simplify both surds separately by finding the highest square number that is a factor of each of them

9 is a factor of 54, so $square root of 54 space equals space square root of 9 space cross times space 6 end root space equals space 3 square root of 6$

4 is a factor of 24, so $square root of 24 space equals space square root of 4 space cross times space 6 end root space equals space 2 square root of 6$

Simplify the whole expression by collecting the like terms

$square root of 54 space minus square root of 24 space equals space 3 square root of 6 space minus 2 square root of 6 space space equals space square root of 6$

$square root of bold 54 bold space bold minus bold space square root of bold 24 bold space bold equals bold space square root of bold 6$