Simplifying Surds (Edexcel IGCSE Further Maths)

Surds & Exact Values

Spec point — 1C

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. $5\sqrt{2}$ rather than $7.071067811...$

How do I calculate with surds?

• Multiplying surds

  • You can multiply numbers under square roots together

  • $\sqrt{a}\times\sqrt{b}=\sqrt{ab}$

    • e.g. $\sqrt{3}\times\sqrt{5}=\sqrt{3\times5}=\sqrt{15}$

• Dividing surds

  • You can divide numbers under square roots

  • $\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$

    • e.g. $\sqrt{21}\div\sqrt{7}=\sqrt{21\div7}=\sqrt{3}$

• Factorising surds

  • You can factorise numbers under square roots

    • This lets you split a single square root into a product of square roots

  • $\sqrt{ab}=\sqrt{a}\times\sqrt{b}$

    • e.g. $\sqrt{35}=\sqrt{5\times7}=\sqrt{5}\times\sqrt{7}$

• Adding or subtracting surds is very like adding or subtracting letters in algebra

  • you can only add or subtract multiples of “like” surds

    • e.g. $3\sqrt{5}+8\sqrt{5}=11\sqrt{5}$ or $7\sqrt{3}--4\sqrt{3}=3\sqrt{3}$
    • but $3\sqrt{5}-4\sqrt{3}$ can't be simplified further

  • Be very careful here, you cannot add or subtract numbers under square roots

    • $\sqrt{a}+\sqrt{b}$ is not equal to $\sqrt{a+b}$
    • $\sqrt{a}-\sqrt{b}$ is not equal to $\sqrt{a-b}$

  • Think about $\sqrt{9}+\sqrt{4}=3+2=5$

    • It is not equal to $\sqrt{9+4}=\sqrt{13}=3.60555\dots$

Exam Hint

• If your calculator gives you an answer as a surd

  • Leave the value as a surd throughout the rest of your calculations
  • This will make sure you do not lose accuracy
  • Round only at the very end (if necessary)

• A question might ask for an 'exact value' answer

  • In that case leave your answer as a surd

Simplifying Surds

Spec point — 1C

How do I simplify surds?

• To simplify a surd, separate out any square factors and take their square root

  • Look for the greatest square number that is a factor of the number you are simplifying

    • e.g. $\sqrt{48}=\sqrt{16\times3}=\sqrt{16}\times\sqrt{3}=4\times\sqrt{3}=4\sqrt{3}$

  • If you don't spot the greatest square factor the first time continue the process

    • e.g. $\sqrt{450}=\sqrt{9\times50}=\sqrt{9}\times\sqrt{50}=3\times\sqrt{50}=3\sqrt{50}$
    • But 50 still has a square factor so continue
    • $3 \sqrt{50 = 3 \sqrt{25} \times 2} = 3 ( \sqrt{25} \times \sqrt{2} ) = 3 ( 5 \times \sqrt{2} ) = 15 \times \sqrt{2} = 15 \sqrt{2}$

• You can collect like terms with surds like you do with letters in algebra

• Understanding how to simplify surds can help with simplifying expressions and collecting like terms

  • e.g. simplify $\sqrt{32}+\sqrt{8}$ by simplifying each part separately

$\begin{gathered} \sqrt{32} + \sqrt{8} & = ( \sqrt{16} \times \sqrt{2} ) + ( \sqrt{4} \times \sqrt{2} ) \\ = 4 \sqrt{2} + 2 \sqrt{2} \\ = 6 \sqrt{2} \end{gathered}$

• An important skill is expanding brackets containing surds

  • This is done in the same way as expanding brackets algebraically
  • But the property $( \sqrt{a} )^{2} = a$ can be used to simplify the expression, once expanded

Exam Hint

• In exam questions different surds being simplified will often have the same non-square factor

  • This can help you find the correct highest square factors

  • e.g. $\sqrt{48}+\sqrt{75}$ can be rewritten as $\sqrt{3\times16}+\sqrt{3\times25}$

    • This then simplifies easily to $4\sqrt{3}+5\sqrt{3}=9\sqrt{3}$

Worked Example Write $\sqrt{54}-\sqrt{24}$ in the form $p\sqrt{q}$ where $p$ and $q$ are integers and $q$ has no square factors.

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

$\sqrt{54}=\sqrt{9\times6}=3\sqrt{6}$

4 is a factor of 24, so

$\sqrt{24}=\sqrt{4\times6}=2\sqrt{6}$

Simplify the whole expression by collecting the like terms

• *$\sqrt{54}-\sqrt{24}=3\sqrt{6}-2\sqrt{6}=\sqrt{6}$

This is in the required form with $p=1$ and $q=6$

$\sqrt{6}$