Grade AA Surds Simplify expressions involving surds including

Grade A/A* Surds Simplify expressions involving surds, including rationalising denominators If you have any questions regarding these resources or come across any errors, please contact helpful-report@pixl. org. uk

Key Vocabulary Simplify Surd Rationalise Denominator

How to simplify expressions involving surds. We are sometimes asked to simplify surds by writing them in the form a√b. Start by finding the largest square number that divides into the surd. √ 32= √(16 x 2) =√ 16 x √ 2 = 4√ 2 √ 50 = √(25 x 2) = √ 25 x √ 2 =5√ 2 √ 300 = √(100 x 3) = √ 100 x √ 3 = 10 √ 3 √ 180 = √(36 x 5) = √ 36 x √ 5 = 6√ 5

How to simplify expressions involving surds We may also need to expand simplify using surds 3(5 - √ 3) = 15 - 3 √ 5 (2+ √ 3) = 2 √ 5 + √ 15 (1 + √ 3) (2 + √ 8) = 2 + √ 8 + 2√ 3+ √ 24 = 2 + √(4 x 2) + 2√ 3 + √(4 x 6) = 2 + √ 4 x √ 2 + 2√ 3 + √ 4 x √ 6 = 2 + 2√ 2 + 2 √ 3 + 2 √ 6

How to simplify expressions with surds including rationalising the denominator Rationalise each expression: X √ 8 X √ 6 = X √ 8 = ÷ 2 Can simplify! = x (1+√ 2) = Multiply by the inverse! Expand the brackets = = = = ÷ 2 = = = -3(1 + √ 2)

Simplify surds – now you try. . . a)√ 75 b)√ 40 c)√ 162 d)√ 117 e) √ 5( 2+ √ 4) f) (√ 3+ 2)(√ 5+3) g) (2 - √ 3)2

Simplify surds – now you try. . . a)√ 75 = 5 √ 3 b)√ 40 = 2 √ 10 c)√ 162 = 9 √ 2 d)√ 117 = 3 √ 13 e) √ 5( 2+ √ 4) =4√ 5 f) (√ 3+ 2)(√ 5+3) = √ 15 + 3√ 3 +2√ 5 + 6 g) (2 - √ 3)2 = 7 - 4 √ 3

Rationalising denominators

Rationalising denominators = = =

Problem Solving and Reasoning Jim wants to put a fence around his rectangular field. The exact length and width of the field are given opposite. a) How long must his fencing be? b) What is the area of the field? (√ 2 + 6) m √ 5 m

Problem Solving and Reasoning Jim wants to put a fence around his rectangular field. The exact length and width of the field are given opposite. a) Exactly how long must his fencing be? 2 √ 5+2 √ 2+ 12 m b) What is the exact area of the field? √ 10 + 6 √ 5 m (√ 2 + 6) m √ 5 m

Reason and explain Why do we refer to the ‘exact value’ with surds ? Explain in words how to Rationalise a denominator. Think also about the ‘inverse operation rule’ What do you look for when you simplifying a surd? Spot the mistake: =