Surds www mathsrevision com S 4 Credit Simplifying

Surds www. mathsrevision. com S 4 Credit Simplifying a Surd Rationalising a Surd www. mathsrevision. com

Starter Questions www. mathsrevision. com S 4 Credit Use a calculator to find the values of : =6 = 12 =2 =2 www. mathsrevision. com

The Laws Of Surds www. mathsrevision. com S 4 Credit Learning Intention 1. To explain what a surd is and to investigate the rules for surds. www. mathsrevision. com Success Criteria 1. Learn rules for surds. 1. Use rules to simplify surds.

Surds S 4 Credit www. mathsrevision. com We can describe numbers by the following sets: N = {natural numbers} = {1, 2, 3, 4, ………. } W = {whole numbers} = {0, 1, 2, 3, ………. . } Z = {integers} = {…. -2, -1, 0, 1, 2, …. . } Q = {rational numbers} This is the set of all numbers which can be written as fractions or ratios. eg 5 = 5/ 1 -7 = 55% = -7/ 55/ 0. 6 = 6/10 = 3/5 1 100 = 11/ 20 etc

Surds www. mathsrevision. com S 4 Credit R = {real numbers} This is all possible numbers. If we plotted values on a number line then each of the previous sets would leave gaps but the set of real numbers would give us a solid line. We should also note that N “fits inside” W W “fits inside” Z Z “fits inside” Q Q “fits inside” R

www. mathsrevision. com Surds N W Z Q R When one set can fit inside another we say that it is a subset of the other. The members of R which are not inside Q are called irrational (Surd) numbers. These cannot be expressed as fractions and include , 2, 3 5 etc

What is a Surd www. mathsrevision. com S 4 Credit =6 = 12 The above roots have exact values and are called rational These roots do NOT have exact values and are called irrational OR www. mathsrevision. com Surds

Note : Adding & Subtracting √ 2 +Surds √ 3 does not equal √ 5 S 4 Credit www. mathsrevision. com Adding and subtracting a surd such as 2. It can be treated in the same way as an “x” variable in algebra. The following examples will illustrate this point. www. mathsrevision. com

First Rule www. mathsrevision. com S 4 Credit Examples List the first 10 square numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 www. mathsrevision. com

Simplifying Square Roots www. mathsrevision. com S 4 Credit Some square roots can be broken down into a mixture of integer values and surds. The following examples will illustrate this idea: 12 = 4 x 3 = 2 3 To simplify 12 we must split 12 into factors with at least one being a square number. Now simplify the square root. www. mathsrevision. com

Have a go ! Think square numbers www. mathsrevision. com S 4 Credit 45 32 72 = 9 x 5 = 16 x 2 = 4 x 18 = 3 5 = 4 2 = 2 x 9 x 2 = 2 x 3 x 2 = 6 2 www. mathsrevision. com

What Goes In The Box ? www. mathsrevision. com S 4 Credit Simplify the following square roots: (1) 20 (2) 27 (3) 48 = 2 5 = 3 3 = 4 3 (4) 75 (5) 4500 (6) 3200 = 5 3 = 30 5 = 40 2 www. mathsrevision. com

First Rule www. mathsrevision. com S 4 Credit Examples www. mathsrevision. com

Have a go ! www. mathsrevision. com S 4 Credit Think square numbers

Have a go ! www. mathsrevision. com S 4 Credit Think square numbers

Exact Values S 4 Credit www. mathsrevision. com Now try MIA Ex 7. 1 Ex 8. 1 Ch 9 (page 185) 25 -Feb-21 Created by Mr Lafferty Maths Dept

Starter Questions www. mathsrevision. com S 4 Credit Simplify : = 2√ 5 =¼ www. mathsrevision. com = 3√ 2 =¼

The Laws Of Surds www. mathsrevision. com S 4 Credit Learning Intention 1. To explain how to rationalise a fractional surd. www. mathsrevision. com Success Criteria 1. Know that √a x √a = a. 2. To be able to rationalise the numerator or denominator of a fractional surd.

Second Rule www. mathsrevision. com S 4 Credit Examples www. mathsrevision. com

Rationalising Surds www. mathsrevision. com S 4 Credit You may recall from your fraction work that the top line of a fraction is the numerator and the bottom line the denominator. Fractions can contain surds: www. mathsrevision. com

Rationalising Surds www. mathsrevision. com S 4 Credit If by using certain maths techniques we remove the surd from either the top or bottom of the fraction then we say we are “rationalising the numerator” or “rationalising the denominator”. Remember the rule This will help us to rationalise a surd fraction www. mathsrevision. com

Rationalising Surds www. mathsrevision. com S 4 Credit To rationalise the denominator multiply the top and bottom of the fraction by the square root you are trying to remove: ( 5 x 5 = 25 = 5 ) www. mathsrevision. com

Rationalising Surds www. mathsrevision. com S 4 Credit Let’s try this one : Remember multiply top and bottom by root you are trying to remove www. mathsrevision. com

Rationalising Surds www. mathsrevision. com S 4 Credit Rationalise the denominator www. mathsrevision. com

What Goes In The Box ? www. mathsrevision. com S 4 Credit Rationalise the denominator of the following : www. mathsrevision. com

Looks something like the difference S 4 Credit of two squares www. mathsrevision. com Rationalising Surds Conjugate Pairs. Look at the expression : This is a conjugate pair. The brackets are identical apart from the sign in each bracket. Multiplying out the brackets we get : = 5 x 5 - 2 5 + 2 5 - 4 =5 -4 =1 When the brackets are multiplied out the surds ALWAYS cancel out and we end up seeing that the expression is rational ( no root sign ) www. mathsrevision. com

Third Rule Conjugate Pairs. www. mathsrevision. com S 4 Credit Examples =7– 3=4 = 11 – 5 = 6 www. mathsrevision. com

Rationalising Surds www. mathsrevision. com S 4 Credit Conjugate Pairs. Rationalise the denominator in the expressions below by multiplying top and bottom by the appropriate conjugate: www. mathsrevision. com

Rationalising Surds www. mathsrevision. com S 4 Credit Conjugate Pairs. Rationalise the denominator in the expressions below by multiplying top and bottom by the appropriate conjugate: www. mathsrevision. com

What Goes In The Box www. mathsrevision. com S 4 Credit Rationalise the denominator in the expressions below : Rationalise the numerator in the expressions below : www. mathsrevision. com

Rationalising Surds S 4 Credit www. mathsrevision. com Now try MIA Ex 9. 1 Ch 9 (page 188) 25 -Feb-21 Created by Mr Lafferty Maths Dept