Simplifying Surds (Edexcel IGCSE Maths)

Surds & Exact Values

Spec point — 1.4B

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 $7.071067812...$

How do I do calculations with surds?

• Multiplying surds

  • You can multiply numbers under square roots together
  • $\sqrt{3}\times\sqrt{5}=\sqrt{3\times5}=\sqrt{15}$

• Dividing surds

  • You can divide numbers under square roots
  • $\frac{\sqrt{21}}{\sqrt{7}}=\sqrt{21}\div\sqrt{7}=\sqrt{21\div7}=\sqrt{3}$

• Factorising surds

  • You can factorise numbers under square roots
  • $\sqrt{35}=\sqrt{5\times7}=\sqrt{5}\times\sqrt{7}$

• Adding or subtracting surds

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

    • This is similar to collecting like terms when simplifying algebra

  • $3\sqrt{5}+8\sqrt{5}=11\sqrt{5}$

  • $7\sqrt{3}--4\sqrt{3}=3\sqrt{3}$

    • However $2\sqrt{3}+4\sqrt{6}$ cannot be simplified

  • You cannot add or subtract numbers under square roots

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

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

Exam Hint If your calculator gives an answer as a surd, leave the value as a surd throughout the rest of your working.

This will ensure you do not lose accuracy throughout your working.

Simplifying Surds

Spec point — 1.4B

How do I simplify surds?

• To simplify a surd, factorise the number using a square number, if possible

  • If multiple square numbers are a factor, use the largest

• Use the fact that $\sqrt{ab}=\sqrt{a}\times\sqrt{b}$ and then work out any square roots of square numbers

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

• When simplifying multiple surds, simplify each separately

  • This may produce surds which can then be collected together

    • E.g. $\sqrt{32}+\sqrt{8}$ can be rewritten as $\sqrt{16}\sqrt{2}+\sqrt{4}\sqrt{2}$
    • This simplifies to $4\sqrt{2}+2\sqrt{2}$
    • These surds can then be collected together
    • $6\sqrt{2}$

• You may have to expand double brackets containing surds

  • This can be done in the same way as multiplying out double brackets algebraically, and then simplifying

  • The property (√(a)²=a can be used to simplify the expression, once expanded

  • E.g. (√(6−2)(√(6+4) expands to (√(6)²+4 √(6−2 √(6−8

    • This simplifies to $6+2\sqrt{6}-8$ which gives $-2+2\sqrt{6}$

Worked Example Write $\sqrt{54}-\sqrt{24}$ in the form $\sqrt{q}$ where $q$ is a positive integer.

Answer:

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

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

$\sqrt{6}$