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Rationalising the Denominator - Revision Notes

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7.5Rationalising the Denominator


In these revision notes for Rationalising the Denominator, we cover the following key points:

  • How to multiply brackets containing surds?
  • How to use special algebraic identities to write expressions containing surds in a simple way?
  • What does 'rationalising the denominator' mean?
  • How to rationalise the denominator of a fraction if it contains only a surd and nothing else?
  • How to rationalise the denominator of a fraction if it contains only a surd multiplied by a number.
  • How to rationalise the denominator of a fraction if it contains a binomial?
  • How to rationalise the denominator when an entire fraction is inside a root?

Rationalising the Denominator Revision Notes

Rationalising the denominator means getting rid of roots when they are in the denominator of a fraction, especially when these roots contain surds.

We may often need the help of the eight special algebraic identities to multiply brackets containing surds. In many cases, they also allow us to rationalise the denominator of a fraction.

There exist three kind of situations in which we may need to rationalise the denominator.

1. The denominator contains a single term with a root

When denominator is made of a single term containing a root, we simply multiply up and down (numerator and denominator) by the root itself as many times as needed, depending on the root index. This eliminates the root from denominator.

2. The denominator contains a binomial where at least one of terms contains a surd

In this case, we multiply up and down by the conjugate of denominator. This allows get rid of the surd in the denominator as it raises at the second power.

3. Rationalising the denominator when an entire fraction is inside a root

When we have the root of an entire fraction involved, the first thing to do is to apply the rule

√(a/b) = √a/√b

where b can also be a binomial, not just a number. Then, we use the any of the first two methods to rationalise the denominator.

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