Understanding Surds in Mathematics

GCSE: Surds Dr J Frost (jfrost@tiffin.kingston.sch.uk) www.drfrostmaths.com Last modified: 19th November 2019

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Types of 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, 31 2, -7.

Types of numbers Activity: Copy out the Venn diagram, and put the following numbers into the correct set. Real numbers Rational numbers 3 0.7 Integers . 1.3 2 -1 3 4 9 e Edwin s exact height (in m) (Click the blue boxes above)

What is a surd? Vote on whether you think the following are surds or not surds. 2 Not a surd Surd 9 Not a surd Surd Not a surd Surd 5 1 4 Not a surd Surd 37 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.

Laws of Surds The only two things you need to know this topic ? ? ? = ?? ? ?= ? ? ? Basic Examples: 1 9=? ?? 3 2 = ? ? ? ??= ?? ? 4?2=

Simplifying Surds ? ? = ? ? ? 8 = ? Bro Tip: Find the largest square factor of the number, and put that first. Could we somehow use ?? = to break the 8 up in a way that one of the surds will simplify? ? ? ? 27 = ? ? = ? ? ? 32 = ?? ? = ? ? ? 2 50 = ? ?? ? = ?? ? 4 12 = ? ? ? = ? ? ?

Test Your Understanding ? 75 = ?? ? = ? ? ? 20 = ? ? = ? ? 48 = ?? ? = ? ? ? 3 200 = ? ??? ? = ?? ? ? 5 45 = ? ? ? = ?? ? ?

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

Test Your Understanding ? 6 7 = ? ? 5 6 = 2 2 = ? ? 5 3 4 3 = ?? 3 4 3 = ?? ? 2= ?? 5 8 2 2 = ?? 2 2 3 6 = ? ?? = ? ? ? = ?? ? ?? ? ? ? ? ? 3 5 ? ?

Exercise 1 Simplify the following: 8 = ? ? 18 = ? ? 50 = ? ? 80 = ? ? 72 = ? ? ? 1 4 a b c d Simplify the following: 8 3 2 = ?? 27 2 3 = ?? 3 18 2 = ?? 2 12 3 3 = ?? ?? ? ? ?? ?? a b c d e Express the following as a single square root (hint: do the steps of simplification backwards!) 3 2 = ? ? = 2 5 = ? ? = 5 7 = ??? 4 3 = ?? ? 5 Simplify the following: 5 80 = ?? ? 2 125 = ?? ? 8 12 = ?? ? 3 72 = ?? ? 2 28 = ? ? 2 ?? ? ? ? a b c d e ? ? ?? ?? a b c d ? 3 Simplify the following: 3 2 5 = 27 3 = ? 4 3 2 = ? ? 5 2 5 = ?? ? 2 2 2 2 = ? 7 3 2 5 = ?? ?? 6 3 2 3 = ?? ?? ? a b c d e f g Express the following as a single square root: ??? 2 ? = ?? ? ? 6 ??? ? ? ? = a b ? ?

Adding Surds 3 + 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. ? + ? = 2? ? 2 5 + 5 = ? ? 7 7 + 7 7 = ?? ? 2 + 8 = = ? ? 3 12 + 27 = ? ? ? + ? ? = ? ? + ? ? = ? ? ? ? + ? ? ? ?

Test Your Understanding ? 3 + 3 + 3 = ? ? 8 + 18 2 5 + 2 20 = ? ? + ? ? 3 48 + 12 = ?? ? + ? ? = ?? ? ? = ? ? + ? ? = ? ? ? = ?? ? ?

Brackets and Surds ? 2 3 + 2 = ? ? + ? ? 2 + 1 2 1 = ? + ? ? ? = ? 8 + 3 2 + 5 = ?? + ? ? + ? ? + ?? = ? + ? ? ? + ? ? + ?? = ?? + ?? ? + ? ? = ?? + ?? ? 2= ? ? = ? ? ? ? ? + ? = ? ? ? ? 5 2 ? ? ?

Test Your Understanding ? 5 2 + 3 1 + 3 8 1 = ? ? + ?? ? 2 + 3 = ? + ? ? 2 + 3 = ? + ? ? 2 = ?? ?? ? ? ? 3 2 5 ???? = ? + ? ? ? 3 + 3 1 + 3 3

Exercise 2 3 a 1 a b c d e f Simplify the following: 8 + 18 = ? ? 12 3 = 20 + 45 = ? ? 3 + 12 + 27 = ? ? 300 48 = ? ? 2 50 + 3 32 = ?? ? Expand and simplify: 2 8 2 + 2 = ? ? 3 + 27 4 3 = ? + ? ? 2 2 + 5 4 + 18 = ?? + ?? ? 8 + 18 32 50 = ?? 2 2 1 2 3 2 Determine the area of : b ?? ? ? ? b c d ? ? ? ? ?? 2= ? ? 2= ? ? ? ? 2 + 1 e 3 + 2 f 2 Expand and simplify the following, leaving your answers in the form ? + ? ? 3 2 + 3 = ? ? + ? 3 + 1 2 + 3 = ? + ? ? 5 1 2 + 5 = ? + ? 7 + 1 2 2 7 = ?? 2= ? ? ? 4 c 3 a 4 + 3 5 5 1 6 5 5 2 + 3 ? a ? ? ? b c d 5 + 3 ? ? ? = ? + ? ? = ? + ? ? ? ? = ? ? + ? ? 2 3 5 e ? Find the length of ??. (Using Pythagoras) ?? = ? 7 2 ?? ? 7 + 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. Bro Side Note: There s two reasons why we might want to do this: 1. 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! 2. It makes it easier for us to add expressions involving surds. What could we multiply this fraction by to both rationalise the denominator, but leave the value of the fraction unchanged?

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

FURTHER MATHS! :: More Complex Denominators 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 = ? ? 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 ? + ? ? ? = ?2 ?2, any surds will be squared and thus we ll end up with no surds in the denominator.

More Examples You can explicitly expand out 6 2 6 + 2 in the denominator, but remember that ? ? ? + ? = ?2 ?2 so we get 6 4 = 2 Just remember: difference of two squares ! 3 6 + 2 6 + 2=3 6 + 6 ? ? 6 2 2 4 3 1 3 1=4 3 4 3 + 1 = 2 3 2 ? 2 ? ? 3 2 + 4 5 2 7 ? ? + ? ? ? + ?=?? + ?? ? + ?? ? + ?? = ?? + ?? ? ? ? ?

Test Your Understanding Rationalise the denominator and simplify 4 ? + ? ? ? 5 2 Rationalise the denominator and simplify 2 3 1 AQA FM June 2013 Paper 1 3 3 + 1 Solve ? Give your answer in the form ? + ? 3 where ? and ? are integers. 3 1 = 8 ? ? ? ? ? + ? ? ? ? =?? ? ? ? ? + ? ?? ? =?? ? ? ?? ? ? ? ? ? ? + ? ? = ? ? ? ? + ? =? ? + ? = ? + ? ? ?

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

A final super hard puzzle 49 527= ?3 Solve ? ? ? ? ??? ???= ?? =? = ? ? ?? ? ? ? ? ?? ? ? ? ? ?? = ? ?= ? But ? ?? ? = ??