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Rationalising surds - Higher

Rationalising surds - Higher

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Explainer Video

Tutor: Toby

Summary

Rationalising surds

​​In a nutshell

Single or double brackets involving surds can be expanded or multiplied out in a similar way to expanding single or double brackets with algebraic expressions. Rationalising surds eliminates the irrational number in the denominator. You can think of this as converting to equivalent fractions with a rational denominator. Surd fractions with more complex denominators will need the conjugate surd to rationalise them.



Definitions

Rationalising
Converting a fraction with an irrational denominator to one with a rational denominator.

Conjugate
A binomial expression with the same terms as the original, with the opposite sign in the middle.



Expanding double brackets with surds

To expand double brackets, you can treat surds as you would algebraic terms. Make sure to multiply each term in the first bracket by each term in the second bracket. You can use the F.O.I.L. method or a multiplication grid.


FOIL

First
Multiply the first terms of each bracket.
Outside
Multiply the outside terms.
Inside
Multiply the two inside terms.
Last
Multiply the last terms of each bracket.

Note: Once the brackets have been multiplied out, simplify the terms where possible.

Maths; Number; KS4 Year 10; Rationalising surds - Higher

​

Example 1

Expand (1+2)(2+3)(1+\sqrt2)(2+\sqrt{3})(1+2​)(2+3​)  using the F.O.I.L. method.


​(1+2)(2+3)=2⏟First+3⏟Outside+22⏟Inside+6⏟Last=2+3+22+6‾\begin{aligned}(1+\sqrt2)(2+\sqrt{3})&= \underbrace{2}_{\text{First}}+\underbrace{\sqrt{3}}_{\text{Outside}}+\underbrace{2\sqrt2}_{\text{Inside}}+\underbrace{\sqrt6}_{\text{Last}} \\&=\underline{2+\sqrt3+2\sqrt2+\sqrt6}\end{aligned}(1+2​)(2+3​)​=First2​​+Outside3​​​+Inside22​​​+Last6​​​=2+3​+22​+6​​​​​

​

MULTIPLICATION GRID

A multiplication grid separates each term of either bracket as headers for rows and columns. The rest of the table is filled with the corresponding products.


Example 2

Expand (4+3)(1−3) (4+\sqrt3)(1-\sqrt3) (4+3​)(1−3​).


Build your multiplication grid.

​×\times×​​​
​444​​
​3\sqrt33​​​
​111​​
​444​​
​3\sqrt33​​​
​−3-\sqrt3−3​​​
​−43-4\sqrt3−43​​​
​−3-3−3​​​


Add the terms and simplify.

​​​(4+3)(1−3)=4+3−43−3=1−33‾\begin {aligned} (4+\sqrt3)(1-\sqrt3) &= 4 + \sqrt3 -4\sqrt3 -3 \\ &= \underline{1-3\sqrt3} \end {aligned} (4+3​)(1−3​)​=4+3​−43​−3=1−33​​​


Example 3

Multiply out and simplify (4+3)(1+3)(4+\sqrt3)(1+\sqrt3)(4+3​)(1+3​).


(4+3)(1+3)=4+43+3+3=7+53‾\begin {aligned} (4+\sqrt3)(1+\sqrt3) &= 4 + 4\sqrt3 +\sqrt3 +3 \\ &= \underline{7+5\sqrt3} \end {aligned} (4+3​)(1+3​)​=4+43​+3​+3=7+53​​​​​



Rationalising the denominator

Rationalising the denominator means removing the surd from the bottom of a fraction, by converting to an equivalent fraction. This is done by multiplying the numerator and the denominator by the same term.


To rationalise a surd, you multiply the numerator and the denominator by the surd in the denominator.


Example 4

Rationalise the denominator 15\dfrac{1}{\sqrt5}5​1​.


Multiply both the numerator and the denominator by the surd in the denominator. Here, it is 5\sqrt55​. So:

15=1×55×5=55‾\begin {aligned} \frac{1}{\sqrt5} &= \frac{1\times \sqrt5}{\sqrt5 \times \sqrt5}\\ &= \underline{\frac{\sqrt5}{5}} \end {aligned} 5​1​​=5​×5​1×5​​=55​​​​​​


Rationalising for more complex denominators

Sometimes you can have more complex denominators like:

12+3\dfrac{1}{2+\sqrt3}2+3​1​​​

Multiplying top and bottom by 3\sqrt33​ will still leave a surd in the denominator. You need the conjugate expression of the denominator to eliminate the surd. In this case, the conjugate is:

 2−32-\sqrt32−3​​


procedure

1.
Form the conjugate of the denominator.
2.
Multiply both the numerator and the denominator by the conjugate.
3.
Simplify the answer.

​

Example 5

Rationalise the denominator 103+2\frac{10}{3+\sqrt2}3+2​10​.


The conjugate is 3−23-\sqrt23−2​.


Multiply both the numerator and the denominator by the conjugate.


103+2=10×(3−2)(3+2)(3−2)=30−1029−32+33−2=30−1027‾\begin {aligned}\frac{10}{3+\sqrt2} &= \frac{10\times (3-\sqrt2)}{(3+\sqrt2)(3-\sqrt2)}\\\\ &=\frac{30-10\sqrt2}{9-3\sqrt2+3\sqrt3-2}\\\\&= \underline{\frac{30-10\sqrt2}{7}} \end {aligned} 3+2​10​​=(3+2​)(3−2​)10×(3−2​)​=9−32​+33​−230−102​​=730−102​​​​​​

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Surds: Simplify, add and subtract - Higher

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FAQs - Frequently Asked Questions

How do you rationalise a surd?

To rationalise a surd, you multiply the numerator and the denominator by the surd in the denominator. For more complex denominators, you need the conjugate expression of the surd.

What does rationalising the denominator mean?

Rationalising the denominator means removing the surd from the bottom of a fraction.

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