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Question
Simplify:
`(7sqrt(3))/(sqrt(10) + sqrt(3)) - (2sqrt(5))/(sqrt(6) + sqrt(5)) - (3sqrt(2))/(sqrt(15) + 3sqrt(2))`
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Solution
`(7sqrt(3))/(sqrt(10) + sqrt(3)) - (2sqrt(5))/(sqrt(6) + sqrt(5)) - (3sqrt(2))/(sqrt(15) + 3sqrt(2))`
Rationalise the denominators:
⇒ `((7sqrt(3))/(sqrt(10) + sqrt(3)) xx (sqrt(10) - sqrt(3))/(sqrt(10) - sqrt(3))) - ((2sqrt(5))/(sqrt(6) + sqrt(3)) xx (sqrt(6) - sqrt(5))/(sqrt(6) - sqrt(5))) - ((3sqrt(2))/(sqrt(15) + 3sqrt(2)) xx (sqrt(15) - 3sqrt(2))/(sqrt(15) - 3sqrt(2)))`
⇒ `(7sqrt(3)(sqrt(10) - sqrt(3)))/(10 - 3) - (2sqrt(5)(sqrt(6) - sqrt(5)))/(6 - 5) - (3sqrt(2)(sqrt(15) - 3sqrt(2)))/(15 - 8)` ...[∵ a2 – b2 = (a + b)(a – b)]
⇒ `(7sqrt(3)(sqrt(10) - sqrt(3)))/(7) - (2sqrt(5)(sqrt(6) - sqrt(5)))/(1) - (3sqrt(2)(sqrt(15) - 3sqrt(2)))/(3)`
⇒ `(7sqrt(30) - 21)/7 - (2sqrt(30) - 10)/1 + (3sqrt(30) - 18)/3`
⇒ `(21sqrt(30) - 63 - 42sqrt(30) + 210 + 21sqrt(30) - 126)/21`
⇒ `21/21 = 1`
Hence the answer is 1.
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