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The Improving Mathematics Education in Schools (TIMES) Project{11}EXAMPLEExpress the following surds with a rational denominator.a2 52 5 – 2 b 3 + 23 2 + 2 3SOLUTIONa2 52 5 – 2 = 2 52 5 – 2 × 2 5 + 22 5 + 2 = 20 + 4 520 – 4 = 5 + 54 .b3 + 23 2 + 2 3 = 3 + 23 2 + 2 3 × 3 2 – 2 33 2 – 2 3 = 3 6 – 6 + 6 – 2 618 – 12= 6 6 .This last example shows quite dramatically how rationalising denominators can, in somecases, simplify a complicated expression to something simpler. However if all that iswanted is an approximation a calculator could be used.EXTENSION-CUBIC SURDSAll of the ideas discussed above can be discussed for surds of the form 3 a.For example:• 5 3 6 + 7 3 6 = 12 36• 2 3 6 × 4 3 36 = 8 363• 103× 10 2= 10LINKS FORWARDMINIMAL POLYNOMIALSSurds arise naturally when solving quadratic and some higher order equations. If we beginwith a quadratic that has integer coefficients and solutions which are surds, then it can beshown that the surds are conjugates of each other. Thus, for example, if we know that acertain quadratic equation with integer coefficients has 2 + 7 as one of its solutions, thenwe can say that the other solution is 2 – 7.Indeed, we can go further and find the monic quadratic equation that has these surds assolutions.Factor the monic quadratic x 2 + bx + c as (x – )(x – ).Expanding this and comparing coefficients gives + = –b, = c.Hence, taking = 2 + 7, = 2 – 7, we haveb = –4, c = (2 + 7)(2 – 7) = –3and so the monic quadratic equation with roots 2 + 7, 2 – 7 isx 2 – 4x – 3 = 0.