Express 4/(2-√2) in the form a+b√2 and write down the values of a and b.

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This is a typical exam question which some students may find confusing.

The trick to this question is realising that you have to rationalise the denominator. (Topic: Surds)

4/(2-√2) = 4/(2-√2) x (2+√2)/(2+√2)

To rationalise, you multiply both the top and the bottom by the conjugate of the denominator. In simple terms, the conjugate is the same as the denominator but with the opposite sign. you are essentially multiplying by 1 so you haven’t actually changed the expression.

= 4(2+√2) / (2-√2)(2+√2)

= (8+4√2) / (4 + 2√2 -2√2 -2)

= (8+4√2) / (4-2)

 As you can see, this has removed the complicated square root from the denominator which makes it easier to simplify.

= 8/2 + (4√2)/2

=4 + 2√2

The rest is simple calculation to get the form the question asks for. So…

a = 4, b = 2

William A. GCSE Maths tutor, A Level Maths tutor

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