Question

How 1/√5+√6-√11=5√6+6√5+√330/60

Answer 1

1/√5+√6-√11 = 1/(√a + √b + √c)

Therefore, 1/[(√a + √b) + √c]  = [(√a + √b) - √c] / { [(√a + √b) + √c].[(√a + √b) - √c] }

= [(√a + √b) - √c] / [(√a + √b)² - c]

= [(√a + √b) - √c] / (a + 2√ab + b - c)

= [(√a + √b) - √c] / [(a + b - c) + 2√ab]

= { [(√a + √b) - √c] * [(a + b - c) - 2√ab] } / { [(a + b - c) + 2√ab] * [(a + b - c) - 2√ab] }

= { [(√a + √b) - √c] * [(a + b - c) - 2√ab] } / [(a + b - c)² - 4ab]

→ in this case: a = 5; b = 6; c = 11

= { [(√5 + √6) - √11] * [(5 + 6 - 11) - 2√30] } / [(5 + 6 - 11)² - 120]

= { [(√5 + √6) - √11] * [- 2√30] } / [- 120]

= (√5 + √6 - √11) * 2√30 / 120

= (√5 + √6 - √11) * √30 / 60

= (√150 + √180 - √330) / 60 → Since: √150 = √(25 * 6) = 5√6

= (5√6 + √180 - √330) / 60 → Since: √180 = √(36 * 5) = 6√5

= (5√6 + 6√5 - √330) / 60 is the solution.

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