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i) Simplify (2 * sqrt(7))^2 ii) Find the value of ((2 * sqrt(7))^2 + 8)/(3 + sqrt(7)) in the form m + n * sqrt(7) where n and m are integers.

i) (2sqrt(7))² = 2² sqrt(7)² = 4 * 7 = 28 (1 mark) ii) First of all substitute the answer from above. ((2 * sqrt(7))² + 8) / (3 + sqrt(7)) = (28 + 8) / (3 + sqrt(7)) = 36 / (3 + sqrt(7)) Multiply by (3 - sqrt(7))/(3 - sqrt(7)) = 1. 36 / (3 + sqrt(7)) = 36 / (3 + sqrt(7)) * (3 - sqrt(7)) / (3 - sqrt(7)) = (36 * (3 - sqrt(7))) / ((3 + sqrt(7)) * (3-sqrt(7)) = (108 - 36 * sqrt(7)) / (9 + 3 * sqrt(7) - 3 * sqrt(7) - 7) = (108 - 36 * sqrt(7)) / 2 = 54 - 18 * sqrt(7). (3 marks) (Hence m=54 and n=-18)

Answered by Andrew B. • Maths tutor

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i) Simplify (2 * sqrt(7))^2 ii) Find the value of ((2 * sqrt(7))^2 + 8)/(3 + sqrt(7)) in the form m + n * sqrt(7) where n and m are integers.

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