Skipton Girls’ High School GCSE: Surds Skipton Girls’ High School

Types of numbers ! Real Numbers Rational Numbers Irrational Numbers Real numbers are any possible decimal or whole number. Real Numbers Rational Numbers Irrational Numbers are real numbers which are not rational. are all numbers which can be expressed as some fraction involving integers (whole numbers), e.g. 1 4 , 3 1 2 , -7.

(Click the blue boxes above) Types of numbers Activity: Copy out the Venn diagram, and put the following numbers into the correct set. Irrational numbers Rational numbers 3 0.7 Integers π . 1.3 √2 -1 3 4 √9 e (Click the blue boxes above)

What is a surd?           ? 2 Not a surd Surd 9 Vote on whether you think the following are surds or not surds. 2 Not a surd   Surd 9 Not a surd  Surd  5  Not a surd Surd  1 4 Not a surd   Surd 3 7  Not a surd Surd  Therefore, can you think of a suitable definition for a surd? A surd is a root of a number that cannot be simplified to a rational number. ?

𝑎 × 𝑏 = 𝒂𝒃 𝑎 𝑏 = 𝒂 𝒃 Laws of Surds ? ? ? ? ? The two main rules you need to know this topic… ? 𝑎 × 𝑏 = 𝒂𝒃 𝑎 𝑏 = 𝒂 𝒃 ? Basic Examples: 1 9 = 𝟏 𝟑 ? ? 3 × 2 = 𝟔 4 𝑥 2 = 𝟒 𝒙 𝟐 =𝟐𝒙 ?

Simplifying Surds Examples ? ? 8 = 𝟒 𝟐 =𝟐 𝟐 Key note: Find the largest square factor of the number, and put that first. Could we somehow use 𝑎𝑏 = 𝑎 𝑏 to break the 8 up in a way that one of the surds will simplify? 27 = 𝟗 𝟑 =𝟑 𝟑 ? 32 = 𝟏𝟔 𝟐 =𝟒 𝟐 ? 2 50 =𝟐 𝟐𝟓 𝟐 =𝟏𝟎 𝟐 ? 4 12 =𝟒 𝟒 𝟑 =𝟖 𝟑 ?

Multiplying Surds Examples 3 × 5 = 𝟏𝟓 2× 3 =𝟐 𝟑 5 ×3=𝟑 𝟓 2 × 8 = 𝟏𝟔 =𝟒 3 × 3 = 𝟗 =𝟑 ? ? Tip: Be very careful in observing whether both of the terms are surds or just one is. ? ? ? Tip: Just multiply the non-surdey things first, then the surdey things. 2 3 ×2 5 =𝟒 𝟏𝟓 ? 3 2 ×3 2 =𝟏𝟖 ? 18 ×4 2 =𝟐𝟒 ?

Exercise 1 75 = 𝟐𝟓 𝟑 =𝟓 𝟑 ? 20 = 𝟒 𝟓 =𝟐 𝟓 ? 48 = 𝟏𝟔 𝟑 =𝟒 𝟑 ? 3 200 =𝟑 𝟏𝟎𝟎 𝟐 =𝟑𝟎 𝟐 ? 5 45 =𝟓 𝟗 𝟓 =𝟏𝟓 𝟓 ?

Exercise 2 6× 7 =𝟔 𝟕 5 × 6 = 𝟑𝟎 2 ×2=𝟐 𝟐 5 3 ×4 3 =𝟔𝟎 3×4 3 =𝟏𝟐 𝟑 3 5 2 =𝟒𝟓 5 8 ×2 2 =𝟒𝟎 2 2 ×3 6 =𝟔 𝟏𝟐 =𝟔 𝟒 𝟑 =𝟏𝟐 𝟑 ? ? ? ? ? ? ? ?

Main Exercise Simplify the following: 8 =𝟐 𝟐 18 =𝟑 𝟐 50 =𝟓 𝟐 80 =𝟒 𝟓 72 =𝟔 𝟐 5 80 =𝟐𝟎 𝟓 2 125 =𝟏𝟎 𝟓 8 12 =𝟏𝟔 𝟑 3 72 =𝟏𝟖 𝟐 2 28 =𝟒 𝟕 3 × 2 × 5 = 𝟑𝟎 27 × 3 =𝟗 4 3 ×2=𝟖 𝟑 5×2 5 =𝟏𝟎 𝟓 2 2 ×2 2 =𝟖 7 3 ×2 5 =𝟏𝟒 𝟏𝟓 6 3 ×2 3 =𝟑𝟔 1 4 Simplify the following: 8 ×3 2 =𝟏𝟐 27 ×2 3 =𝟏𝟖 3 18 × 2 =𝟏𝟖 2 12 ×3 3 =𝟑𝟔 Express the following as a single square root (hint: do the steps of simplification backwards!) 3 2 = 𝟗 𝟐 = 𝟏𝟖 2 5 = 𝟒 𝟓 = 𝟐𝟎 5 7 = 𝟏𝟕𝟓 4 3 = 𝟒𝟖 Express the following as a single square root: 𝑎 𝑏 = 𝒂 𝟐 𝒃 2 𝑘 = 𝟒 𝒌 a ? ? a ? b b ? c ? c ? d ? d ? e ? 5 2 a ? b ? ? c ? a ? d ? b ? ? e c ? 3 d ? a 6 b ? c ? ? a d ? ? b e ? f ? g ?

3 + 3 =𝟐 𝟑 Adding Surds Examples ? 3 + 3 =𝟐 𝟑 ? Think of it as “if I have one lot of 3 and I add another lot of 3 , I have two lots of 3 ”. It’s just how we collect like terms in algebra, e.g. 𝑥+𝑥=2𝑥 2 5 + 5 =𝟑 𝟓 7 7 +7 7 =𝟏𝟒 𝟕 2 + 8 = 𝟐 +𝟐 𝟐 =𝟑 𝟐 3 12 + 27 =𝟑 𝟒 𝟑 + 𝟗 𝟑 =𝟔 𝟑 +𝟑 𝟑 =𝟗 𝟑 ? ? ? ?

Questions asked in books 3 + 3 + 3 =𝟑 𝟑 8 + 18 =𝟐 𝟐 +𝟑 𝟐 =𝟓 𝟐 2 5 +2 20 =𝟐 𝟓 +𝟖 𝟓 =𝟏𝟎 𝟓 3 48 + 12 =𝟏𝟐 𝟑 +𝟐 𝟑 =𝟏𝟒 𝟑 2 5 + 5 =𝟑 𝟓 7 7 +7 7 =𝟏𝟒 𝟕 2 + 8 = 𝟐 +𝟐 𝟐 =𝟑 𝟐 3 12 + 27 =𝟑 𝟒 𝟑 + 𝟗 𝟑 =𝟔 𝟑 +𝟑 𝟑 =𝟗 𝟑

Brackets and Surds Examples 2 3+ 2 =𝟑 𝟐 +𝟐 ? 2 +1 2 −1 =𝟐+ 𝟐 − 𝟐 −𝟏=𝟏 ? 8 +3 2 +5 = 𝟏𝟔 +𝟓 𝟖 +𝟑 𝟐 +𝟏𝟓 =𝟒+𝟓 𝟒 𝟐 +𝟑 𝟐 +𝟏𝟓 =𝟏𝟗+𝟏𝟎 𝟐 +𝟑 𝟐 =𝟏𝟗+𝟏𝟑 𝟐 ? 5 −2 2 = 𝟓 −𝟐 𝟓 −𝟐 =𝟓−𝟐 𝟓 −𝟐 𝟓 +𝟒 =𝟗−𝟒 𝟓 ?

Exercise 1 ? 5 2+ 3 =𝟐 𝟓 + 𝟏𝟓 1+ 3 2+ 3 =𝟓+𝟑 𝟑 8 −1 2 +3 =𝟏+𝟓 𝟐 3−2 5 2 =𝟐𝟗−𝟏𝟐 𝟓 ? ? ? 𝐴𝑟𝑒𝑎=𝟔+𝟓 𝟑 ? 3+ 3 1+3 3

Exercise 2 1 Simplify the following: 8 + 18 =𝟓 𝟐 12 − 3 = 𝟑 20 + 45 =𝟓 𝟓 3 + 12 + 27 =𝟔 𝟑 300 − 48 =𝟔 𝟑 2 50 +3 32 =𝟐𝟐 𝟐 Expand and simplify the following, leaving your answers in the form 𝑎+𝑏 𝑐 3 2+ 3 =𝟐 𝟑 +𝟑 3 +1 2+ 3 =𝟓+𝟑 𝟑 5 −1 2+ 5 =𝟑+ 𝟓 7 +1 2−2 7 =−𝟏𝟐 2− 3 2 =𝟕−𝟒 𝟑 3 Expand and simplify: 2− 8 2+ 2 =−𝟐 𝟐 3+ 27 4− 3 =𝟑+𝟗 𝟑 2 2 +5 4+ 18 =𝟑𝟐+𝟐𝟑 𝟑 8 + 18 32 − 50 =−𝟏𝟎 2 +1 2 − 2 −1 2 =𝟒 𝟐 3 +2 2 − 3 −2 2 =𝟖 𝟑 Determine the area of : ? a ? a b ? ? b c ? ? c d ? ? d e ? ? e f ? ? f 2 4 3 b c a 5 5 −1 4+3 5 6− 5 ? a 2+ 3 ? b 5 +3 ? c 𝑨=𝟏+ 𝟓 𝑨=𝟐 𝟑 +𝟑 ? 𝑨=𝟓+𝟓 𝟓 ? ? ? d ? e N 𝑃 ? Find the length of 𝑃𝑄. (Using Pythagoras) 𝑷𝑸= 𝟐𝟐 7 −2 ? 𝑄 7 +2

5 × 5 =5 1 2 × 2 2 = 2 2 Rationalising The Denominator ? ? ? ? Here’s a surd. What could we multiply it by such that it’s no longer an irrational number? 5 × 5 =5 ? ? 1 2 × 2 2 = 2 2 ? ? In this fraction, the denominator is irrational. ‘Rationalising the denominator’ means making the denominator a rational number. What could we multiply this fraction by to both rationalise the denominator, but leave the value of the fraction unchanged? Note: There’s two reasons why we might want to do this: For aesthetic reasons, it makes more sense to say “half of root 2” rather than “one root two-th of 1”. It’s nice to divide by something whole! It makes it easier for us to add expressions involving surds.

More Examples 3 2 = 𝟑 𝟐 𝟐 6 3 = 𝟔 𝟑 𝟑 =𝟐 𝟑 8 10 = 𝟖 𝟏𝟎 𝟏𝟎 = 𝟒 𝟏𝟎 𝟓 10 5 = 𝟏𝟎 𝟓 𝟓 =𝟐 𝟓 7 7 = 𝟕 𝟕 𝟕 = 𝟕 ? Test Your Understanding: 12 3 =𝟒 𝟑 2 6 = 𝟔 𝟑 4 2 8 = 𝟏𝟔 𝟖 =𝟐 ? ? ? ? ? ? ?

Level 7+ Questions 1 2 +1 × 𝟐 −𝟏 𝟐 −𝟏 = 𝟐 −𝟏 𝟏 = 𝟐 −𝟏 ? ? You’ve seen ‘rationalising a denominator’, the idea being that we don’t like to divide things by an irrational number. But what do we multiply the top and bottom by if we have a more complicated denominator? 1 2 +1 × 𝟐 −𝟏 𝟐 −𝟏 = 𝟐 −𝟏 𝟏 = 𝟐 −𝟏 ? ? We basically do the same but with the sign reversed (this is known as the ‘conjugate’). That way, we obtain the difference of two squares. Since 𝑎+𝑏 𝑎−𝑏 = 𝑎 2 − 𝑏 2 , any surds will be squared and thus we’ll end up with no surds in the denominator.

More Examples 3 6 −2 × 6 +2 6 +2 = 3 6 +6 2 ? ? You can explicitly expand out 6 −2 6 +2 in the denominator, but remember that 𝑎−𝑏 𝑎+𝑏 = 𝑎 2 − 𝑏 2 so we get 6−4=2 Just remember: ‘difference of two squares’! 4 3 +1 × 3 −1 3 −1 = 4 3 −4 2 =2 3 −2 ? ? ? 3 2 +4 5 2 −7 × 𝟓 𝟐 +𝟕 𝟓 𝟐 +𝟕 = 𝟑𝟎+𝟐𝟏 𝟐 +𝟐𝟎 𝟐 +𝟐𝟖 𝟏 =𝟓𝟖+𝟒𝟏 𝟐 ? ?

Test Your Understanding Rationalise the denominator and simplify 4 5 −2 ? 𝟖+𝟒 𝟓 Rationalise the denominator and simplify 2 3 −1 3 3 +1 AQA FM June 2013 Paper 1 Solve 𝑦 3 −1 =8 Give your answer in the form 𝑎+𝑏 3 where 𝑎 and 𝑏 are integers. ? 𝟐 𝟑 −𝟏 𝟑 𝟑 +𝟏 × 𝟑 𝟑 −𝟏 𝟑 𝟑 −𝟏 = 𝟏𝟖−𝟐 𝟑 −𝟑 𝟑 +𝟏 𝟐𝟕−𝟏 = 𝟏𝟗−𝟓 𝟑 𝟐𝟔 ? 𝒚= 𝟖 𝟑 −𝟏 × 𝟑 +𝟏 𝟑 +𝟏 = 𝟖 𝟑 +𝟖 𝟐 =𝟒+𝟒 𝟑

Exercise 1 Rationalise the denominator and simplify the following: 1 5 +2 = 𝟓 −𝟐 3 3 −1 = 𝟑+ 𝟑 𝟐 5 +1 5 −2 =𝟕+𝟑 𝟓 2 3 −1 3 3 +4 =𝟐− 𝟑 5 5 −2 2 5 −3 =𝟒+ 𝟓 1 2 Expand and simplify: 5 +3 5 −2 5 +1 =𝟒 Rationalise the denominator, giving your answer in the form 𝑎+𝑏 3 . 3 3 +7 3 3 −5 =𝟑𝟏+𝟏𝟖 𝟑 Solve 𝑥 4− 6 =10 giving your answer in the form 𝑎+𝑏 6 . 𝑥= 10 4− 6 =𝟒+ 𝟔 Solve 𝑦 1+ 2 − 2 =3 𝑦= 3+ 2 1+ 2 =𝟐 𝟐 −𝟏 Simplify: 𝑎+1 − 𝑎 𝑎+1 + 𝑎 =𝟐𝒂+𝟏−𝟐 𝒂 𝒂+𝟏 ? ? a 3 ? b ? ? c N ? ? d ? N e ? N ?

A final super hard puzzle Solve 4 9 5 27 = 𝑥 3 N 𝟒 𝟑 𝟐 𝟓 𝟑 𝟑 = 𝟑 𝟐 𝟏 𝟒 𝟑 𝟑 𝟏 𝟓 = 𝟑 𝟏 𝟐 𝟑 𝟑 𝟓 = 𝟑 − 𝟏 𝟏𝟎 But 𝒙 𝟑 = 𝟑 𝟏 𝒙 ∴ 𝟏 𝒙 =− 𝟏 𝟏𝟎 → 𝒙=−𝟏𝟎 ?