Surds

Skill 1: Multiplying Surds 

When multiplying surds you simply multiply the numbers inside the square root.

$sqrt{textcolor{red}{a}} times sqrt{textcolor{blue}{b}} = sqrt{textcolor{red}{a}times textcolor{blue}{b}}$

Example: 

$sqrt{7} times sqrt{2} = sqrt{7 times 2} = sqrt{14}$

$2sqrt{2} times 3sqrt{5} = 2times 3 times sqrt{2times5} = 6sqrt{10}$

$sqrt{6}^2 = sqrt{6} times sqrt{6} = sqrt{6times6} =sqrt{36} = 6$

Skill 2: Dividing Surds 

When dividing surds you simply divide the numbers inside the square root.

$dfrac{sqrt{textcolor{red}{a}}}{sqrt{textcolor{blue}{b}}} = sqrt{dfrac{textcolor{red}{a}}{textcolor{blue}{b}}}$

Example: 

$dfrac{sqrt{10}}{sqrt{5}} = sqrt{dfrac{10}{5}} = sqrt{2}$

$dfrac{8sqrt{12}}{2sqrt{3}} = dfrac{8}{2}timessqrt{dfrac{12}{3}} = 4timessqrt{4} = 4times 2 = 8$

Skill 3: Adding and Subtracting Surds

It is only possible to add and subtract “like” surds, this is similar to collecting like terms

$sqrt{a} + sqrt{a} = 2sqrt{a}$

$5sqrt{b} – 2sqrt{b} = 3sqrt{b}$

Do NOT do this:

$xcancel{sqrt{a} + sqrt{b} = sqrt{a+b}}$

Skill 4: Simplifying Surds

Surds can be simplified if the number within the surd has a square number as one of its factors.

Example: Write $sqrt{28}$ in simplified surd form.

We need need to think of a square number which is a factor of $28$.

$28 = textcolor{red}{4} times 7$

$sqrt{28}=sqrt{4times7}=sqrt{textcolor{red}{4}}timessqrt{7}$

We know that $sqrt{textcolor{red}{4}} =  textcolor{red}{2}$

$sqrt{28}=textcolor{red}{2}timessqrt{7}=textcolor{red}{2}sqrt{7}$

Skill 5: Double brackets and surds 

We can multiply out double brackets containing surds the same way as for quadratics using FOIL, then collect like terms.

$(m+sqrt{n}) (m+sqrt{n})=textcolor{red}{m^2}+textcolor{limegreen}{msqrt{n}}+textcolor{purple}{msqrt{n}}+textcolor{blue}{n} = textcolor{red}{m^2}+textcolor{maroon}{2msqrt{n}}+textcolor{blue}{n}$

Example: 

$begin{aligned} &(sqrt{10} + sqrt{3})(sqrt{10} – sqrt{3}) &= sqrt{10}^2 – sqrt{3}sqrt{10} + sqrt{3}sqrt{10} – sqrt{3}^2   &= 10 -sqrt{30} + sqrt{30} -3 &= 10 – 3   &= 7 end{aligned}$

Skill 6: Rationalise the denominator – Simple

Rationalising the denominator just means removing the surd from the bottom of a fraction. There are two types of question you may encounter, one harder then the other. The first type is shown below.

Example: Rationalise the denominator of the following fraction $dfrac{textcolor{red}{a}}{sqrt{textcolor{blue}{b}}}$

Simply multiply the top and bottom of the fraction by the denominator of the fraction.

$dfrac{textcolor{red}{a}}{sqrt{textcolor{blue}{b}}}  = dfrac{textcolor{red}{a}}{sqrt{textcolor{blue}{b}}} times dfrac{sqrt{textcolor{blue}{b}}}{sqrt{textcolor{blue}{b}}} = dfrac{textcolor{red}{a}sqrt{textcolor{blue}{b}}}{textcolor{blue}{b}}$

Skill 7: Rationalise the denominator – Harder

Rationalising the denominator when there are other terms as well as the surd can be much more tricky.

Example: Rationalise the denominator of the following fraction $dfrac{textcolor{red}{5}}{textcolor{blue}{3+sqrt{5}}}$

Multiply the top and the bottom of the fraction by the denominator with the sign changed. $+$ becomes $–$ and $–$ becomes $+$.

$begin{aligned} dfrac{textcolor{red}{5}}{textcolor{blue}{3+sqrt{5}}}   &= dfrac{textcolor{red}{5}}{textcolor{blue}{3+sqrt{5}}} times dfrac{textcolor{limegreen}{3-sqrt{5}}}{textcolor{limegreen}{3-sqrt{5}}}  &= dfrac{textcolor{red}{5}textcolor{limegreen}{(3-sqrt{5})}}{textcolor{blue}{(3+sqrt{5})}textcolor{limegreen}{(3-sqrt{5})}}   &= dfrac{15-5sqrt{5}}{9-3sqrt{5} + 3sqrt{5} -5}   &= dfrac{15-5sqrt{5}}{9 – 5}   &= dfrac{15-5sqrt{5}}{4} end{aligned}$