Rationalising the denominator
Rationalising an expression means getting rid of any surds from the bottom (denominator) of fractions.
Usually when you are asked to simplify an expression it means you should also rationalise it.
Example
Simplify \(\frac{4}{{\sqrt 3 }}\)
Answer
To rationalise the denominator, multiply the fraction by \(\frac{{\sqrt 3 }}{{\sqrt 3 }}\)
\(= \frac{4}{{\sqrt 3 }} \times \frac{{\sqrt 3 }}{{\sqrt 3 }}\)
\(= \frac{4\sqrt3}{3}\) (Remember \(\sqrt 3 \times \sqrt 3 = 3\))
Now try the question below.
Question
Express \(\frac{5}{\sqrt 7}\) with a rational denominator
\(= \frac{5}{\sqrt 7} \times \frac{\sqrt7}{\sqrt7}\)
\(= \frac{5\sqrt 7}{7}\)
Sometimes the denominator might be more complicated and include other numbers as well as the surd.
If this is the case you need to multiply the fraction by a number that will cancel out the surd. Remember to multiply the numerator by the same number or you will change the value of the fraction.
Example
Express \(\frac{12}{4+\sqrt 7}\) with a rational denominator
Answer
To rationalise the denominator, multiply the fraction by \(\frac{4-{\sqrt7}}{4-{\sqrt7}}\)
\(= \frac{12}{4+\sqrt 7} \,\times\, \frac{4-{\sqrt7}}{4-{\sqrt7}}\)
\(=\frac{12(4-\sqrt7)}{(4+\sqrt7)(4-\sqrt7)}\)
Simplify the denominator by multiplying out the brackets to give \(16-4\sqrt7+4\sqrt7-7\). The \(-4\sqrt7\) and \(+4\sqrt7\) cancel out. This leaves the denominator as \(9\).
\(=\frac{12(4-\sqrt7)}{9}\)
Sometimes we can now divide top and bottom by a common factor. In this case \(3\).
\(=\frac{4(4-\sqrt7)}{3}\)
Now try the question below.
Question
Rationalise \(\frac{2}{{3 - \sqrt 5 }}\)
To rationalise the denominator, multiply the fraction by \(\frac{{3 + \sqrt 5 }}{{3 + \sqrt 5 }}\)
\(= \frac{2}{{3 - \sqrt 5 }} \times \frac{{3 + \sqrt 5 }}{{3 + \sqrt 5 }}\)
\(= \frac{{2(3 + \sqrt 5 )}}{{(3 - \sqrt 5 )(3 + \sqrt 5 )}}\)
Simplify the denominator by multiplying out the brackets to give \(9 +3\sqrt5 -3\sqrt5 -5\) which is \(4\).
\(= \frac{{2(3 + \sqrt 5 )}}{4}\)
\(= \frac{{3 + \sqrt 5 }}{2}\)