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Welcome to GCSE Edexcel Maths revision.

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Topic M 6: Surds and exact calculations.

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This video covers Higher tier.

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A surd is an irrational root left in exact form, such as square root of 2. square root of 9 equals 3 is rational and is not a surd.

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An irrational decimal neither terminates nor repeats.

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For non-negative a and b, square root of open bracket a b close bracket equals square root of a multiplied by square root of b.

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Remove square factors: square root of 72 equals square root of open bracket 36 multiplied by 2 close bracket equals 6 multiplied by square root of 2.

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The sum rule is different: square root of open bracket a plus b close bracket is generally not square root of a plus square root of b.

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Like surds combine like algebraic terms: 3 multiplied by square root of 5 plus 2 multiplied by square root of 5 equals 5 multiplied by square root of 5,

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but square root of 2 plus square root of 3 cannot be collected into one root.

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Worked example: square root of 48 plus square root of 27 equals 4 multiplied by square root of 3 plus 3 multiplied by square root of 3 equals 7 multiplied by square root of 3.

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Simplify the roots before looking for like terms.

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Multiplying surds can give a rational result: square root of 3 multiplied by square root of 12 equals square root of 36 equals 6.

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Keep exact values until the final rounding, if rounding is requested.

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Expand surd brackets term by term: open bracket square root of 3 plus 2 close bracket multiplied by open bracket square root of 3 minus 1 close bracket equals 3 minus square root of 3 plus 2 multiplied by square root of 3 minus 2 equals 1 plus square root of 3.

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Conjugates have opposite signs: open bracket a plus square root of b close bracket multiplied by open bracket a minus square root of b close bracket equals a squared minus b.

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The middle surd terms cancel.

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Rationalising removes a surd from a denominator by multiplying numerator and denominator by the same suitable expression.

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This leaves the fraction's value unchanged.

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Worked example: the fraction with numerator open bracket 5 close bracket and denominator open bracket square root of 2 close bracket ,

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end fraction equals the fraction with numerator open bracket 5 multiplied by square root of 2 close bracket and denominator open bracket 2 close bracket ,

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end fraction .

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Multiplying only the denominator would change the value.

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Worked example: the fraction with numerator open bracket 3 close bracket and denominator open bracket 2 plus square root of 3 close bracket ,

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end fraction equals the fraction with numerator open bracket 3 multiplied by open bracket 2 minus square root of 3 close bracket close bracket and denominator open bracket open bracket 2 plus square root of 3 close bracket multiplied by open bracket 2 minus square root of 3 close bracket close bracket ,

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end fraction equals 6 minus 3 multiplied by square root of 3,

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since the denominator becomes 4 minus 3 equals 1.

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If a question asks for an exact answer, use a simplified surd, fraction or multiple of pi as appropriate. 1.414 is an approximation to square root of 2, not the same exact number.

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For a square with area 18 square centimetres ,

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side length is square root of 18 equals 3 multiplied by square root of 2 centimetres and perimeter is 12 multiplied by square root of 2 centimetres.

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Area and length have different units.

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That completes Surds and exact calculations.

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Revisit the notes and test yourself on the revision website.