Surds Simplifying a Surd Rationalising a Surd

Starter Questions 1. = 6 2. = 12 4. = 2 3. = 2 5. 6. =2.76 =1.41 Use a calculator to find the values of : 1. = 6 2. = 12 4. = 2 3. = 2 5. 6. =2.76 =1.41

We can describe numbers by the following sets: Surds We can describe numbers by the following sets: N = {natural numbers} = {1, 2, 3, 4, ……….} W = {whole numbers} = {0, 1, 2, 3, ………..} I = {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 = -7/1 0.6 = 6/10 = 3/5 55% = 55/100 = 11/20 etc

We should also note that Surds 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

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 π ,√2, 3√5 etc

What is a Surd? = 12 = 6 Surds The above roots have exact values and are called rational These roots do NOT have exact values and are called irrational OR Surds

Adding & Subtracting Surds Note : √2 + √3 does not equal √5 Adding & Subtracting Surds 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.

First Rule List the first 10 square numbers Examples List the first 10 square numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, 100

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

Have a go √ 45 √ 32 √ 72 = √9 x √5 = √16 x √2 = √4 x √18 = 3√5 = 4√2 Think square numbers √ 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 10

What Goes In The Box ? Simplify the following square roots: (2) √ 27 (3) √ 48 (1) √ 20 = 2√5 = 3√3 = 4√3 (6) √ 3200 (4) √ 75 (5) √ 4500 = 5√3 = 30√5 = 40√2 11

First Rule Examples

Have a go Think square numbers 13

Have a go Think square numbers

Exact Values

Starter Questions Simplify : 1. 2. = 2√5 = 3√2 3. 4. = ¼ = ¼

Second Rule Examples

Rationalising Surds 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:

Rationalising Surds This will help us to rationalise a surd fraction 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

Rationalising Surds 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 )

Rationalising Surds Let’s try this one : Remember multiply top and bottom by root you are trying to remove

Rationalising Surds Rationalise the denominator

What Goes In The Box ? Rationalise the denominator of the following :

Looks something like the difference of two squares 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 )

Third Rule Conjugate Pairs. Examples = 7 – 3 = 4 = 11 – 5 = 6

Rationalising Surds Conjugate Pairs. Rationalise the denominator in the expressions below by multiplying top and bottom by the appropriate conjugate:

Rationalising Surds Conjugate Pairs. Rationalise the denominator in the expressions below by multiplying top and bottom by the appropriate conjugate:

What Goes In The Box? Rationalise the denominator in the expressions below : Rationalise the numerator in the expressions below :