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Click to start www.maths4scotland.co.uk Higher Mathematics Surds Next Surds Higher Mathematics What are Surds A surd is a square root which cannot be evaluated without approximation. Surds Higher Mathematics What are Surds A surd is an irrational number An irrational number cannot be written as a simple fraction i.e. as a b It is a non-recurring decimal Surds Higher Mathematics Why are surds of interest A surd is a square root which cannot be evaluated without approximation. These often occur when using Pythagoras’ Theorem Trigonometry Using surds allows us to be EXACT Surds Higher Mathematics Examples of Surds 2 3 5 6 etc. A good rule of thumb is to think of a surd as: Any root that cannot be evaluated as a whole number. For example: 4 is not a surd Surds Higher Mathematics Which of these are surds ? 7 20 Yes Yes 25 No 32 81 Yes No Surds Higher Mathematics Rules of surds We can add or subtract surds if they are the same. 7 7 2 7 7 54 5 3 5 Just like algebra: 3x 2 x 5 x Surds Higher Mathematics Examples Simplify 35 32 3 7 52 2 3 5 Surds Higher Mathematics Rules of surds We can multiply surds 5 3 15 Let’s check this out with simple numbers 9 4 3 2 36 6 Surds Higher Mathematics Rules of surds We can multiply surds in general a b ab Note that this works both ways ab a b Surds Higher Mathematics WARNING a b ab Let’s check this out with simple numbers 9 4 13 3 2 5 13 Surds Higher Mathematics Examples Simplify 7 3 8 4 5 7 Surds Higher Mathematics Rules of surds We can divide surds 5 2 5 2 Let’s check this out with simple numbers 6 2 3 36 9 36 9 4 2 Surds Higher Mathematics Rules of surds We can divide surds in general a b a b Note that this works both ways a b a b Surds Higher Mathematics A useful tip a a a a 2 a a 2 a Surds Higher Mathematics Examples Simplify 20 4 24 6 Surds Higher Mathematics Rules a b a b ab a b Surds Higher Mathematics Applications We can use the rules of surds for simplification Simplify: Look for largest square factor 20 45 4 5 2 5 Surds Higher Mathematics Applications We can use the rules of surds for simplification Simplify: Look for largest square factor 50 25 2 25 2 5 2 Surds Higher Mathematics Applications We can use the rules of surds for simplification Simplify: Look for largest square factor 18 2 92 2 9 2 3 2 2 2 2 2 Surds Higher Mathematics Applications We can use the rules of surds for simplification Simplify: Look for largest square factor 75 12 25 3 4 3 25 3 4 3 5 32 3 3 3 Surds Higher Mathematics Applications We can use the rules of surds for simplification Simplify: 3 15 3 15 45 95 9 5 3 5 Surds Higher Mathematics Applications We can use the rules of surds for simplification Simplify: Use FOIL 2 2 3 2 62 2 3 2 2 2 65 2 2 85 2 Surds Higher Mathematics Applications We can use the rules of surds for simplification Simplify: 1 1 2 2 2 2 Use FOIL 1 1 1 2 2 2 2 1 2 2 2 2 1 2 4 2 1 2 2 2 3 2 Surds Higher Mathematics Applications Rationalise the denominator and simplify where possible: 2 6 To get rid the surd – multiply top and bottom by the surd 2 6 6 6 2 6 6 6 3 Surds Higher Mathematics Applications Rationalise the denominator and simplify where possible: 20 5 To get rid the surd – multiply top and bottom by the surd 20 5 5 5 20 5 5 4 5 Surds Higher Mathematics Applications Rationalise the denominator and simplify where possible: 1 2 3 To get rid the surd – multiply top and bottom by the conjugate conjugate - the same expression with the opposite sign in the middle 1 2 3 2 3 2 3 2 3 2 3 2 3 2 3 42 3 2 3 3 3 2 3 Surds Higher Mathematics Applications Rationalise the denominator and simplify where possible: 4 5 1 To get rid the surd – multiply top and bottom by the conjugate conjugate - the same expression with the opposite sign in the middle 4 5 1 5 1 5 1 4 5 1 5 1 4 5 1 4 1 5 www.maths4scotland.co.uk © CPD 2004 Quit THE END