Reducing Surds Revision Notes for Leaving Cert Mathematics

Reducing Surds

In though surds are real they can be treated like independent terms. For example :

\sqrt{11}-2\sqrt{11}+4\sqrt{11}=3\sqrt{11}

You can apply the properties of surds to further simplify :

Example

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Simplify the following expression

a\sqrt{ay^3}+y\sqrt{a^3y}+\sqrt{a^3y^3}

You can use the property $\sqrt{ab}=\sqrt{a} \cdot \sqrt{b}$ to simplify each surd expression. Notice that all the expression under the surd have a common factor of $ay$ .

\begin{align*} a\sqrt{ay(y^2)}+y\sqrt{ay(a^2)}+\sqrt{ay(a^2y^2)} \end{align*}

Apply the property :

\begin{align*} a\sqrt{\left(ay\right)}\sqrt{\left(y^2\right)} +y\sqrt{\left(ay\right)}\sqrt{\left(a^2\right)}+\sqrt{\left(ay\right)}\sqrt{\left(a^2y^2\right)} \end{align*}

Furthermore, you can simplify $\sqrt{a^2y^2}$ :

\begin{align*} a\sqrt{\left(ay\right)}\sqrt{\left(y^2\right)} +y\sqrt{\left(ay\right)}\sqrt{\left(a^2\right)}+\sqrt{\left(ay\right)}\sqrt{\left(a^2\right)}\sqrt{\left(y^2\right)} \end{align*}

Remember that $\sqrt{x^2}=x$ :

\begin{align*} a\sqrt{\left(ay\right)}\cdot y +y\sqrt{\left(ay\right)}\cdot a+\sqrt{\left(ay\right)}\cdot a\cdot y \end{align*}

Since multiplication is commutative, we can rearrange :

\begin{align*} ay\sqrt{ay} +ay\sqrt{ay}+ay\sqrt{ay} \end{align*}

Add like terms :

\begin{align*} 3ay\sqrt{ay} \end{align*}

Rationalising the Denominator

When dividing by a surd, your answer should never have a surd as the denominator. We need to rationalise the denominator.

To rationalise a surd of the form $\sqrt{a}$ , multiply the numerator and denominator by $\sqrt{a}$ .
To rationalise a surd of the form $b\sqrt{a}$ , multiply the numerator and denominator by $\sqrt{a}$ .
To rationalise a surd of the form $c+b\sqrt{a}$ , multiply the numerator and denominator by $c-b\sqrt{a}$ (the conjugate).
To rationalise a surd of the form $a\sqrt{b}+c\sqrt{d}$ , multiply the numerator and denominator by $a\sqrt{b}-c\sqrt{d}$ (the conjugate).

Example

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Rationalise the denominator of the following expression

\frac{1}{\sqrt{5}}

\frac{1}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}=\frac{\sqrt{5}}{5}

Example

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Rationalise the denominator of the following expression

\frac{\sqrt{6}}{2\sqrt{7}}

\frac{\sqrt{6}}{2\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}=\frac{\sqrt{42}}{14}

Example

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Rationalise the denominator of the following expression

\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}

\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}} \cdot \frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}+\sqrt{2}}=\frac{5+2\sqrt{6}}{1}=5+2\sqrt{6}

Reducing Surds (Leaving Cert Mathematics): Revision Notes

Reducing Surds

Rationalising the Denominator

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