Question

Evaluate
$( \sqrt{5 + 2 \sqrt{6)} } + ( \sqrt{8 - 2 \sqrt{15)} }$
​

Answer 1

Answer:

√2 + √5

Explanation:

Simplify each term separately:

Simplify first term:

$\rm \bold{\sqrt{5 + 2\sqrt{6} } }$

By identity: { √a + √b )² = ( a + b + 2√ab )

If 5 + 2√6 is a perfect square, then two numbers a and b should exist, such that  a + b = 5, and ab = √6

By error and trial, we get a and b as 2 and 3

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$\rm\implies\sqrt{2+3+2\sqrt{2\times3}}$

$\rm\implies\sqrt{2+3+2\sqrt{2}\sqrt{3}}$

$\rm\implies\sqrt{\sqrt{(2}) ^2 + 2\sqrt{2} \sqrt{3} +\sqrt{(3)}^2 }$

$\bold{By\:identity: ( a + b)^2 = a^2 + 2ab + b^2}$

$\rm\implies\sqrt{(\sqrt{3}+\sqrt{2}} ^2$

$\rm\implies \sqrt{3} +\sqrt{2}$

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Simplify second term:

$\rm \bold{\sqrt{8 - 2\sqrt{15} } }$

Find a and b , such that a + b = 8, and ab = 15. By error and trial we get a and b as 5 and 3.

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$\rm\implies \sqrt{5+3- 2\sqrt{5}\sqrt{3} }$

$\rm\implies \sqrt{\sqrt{5}^2-2\sqrt{5} \sqrt{3}+\sqrt{3}^2 }$

$\bold {By\: identity: (a-b)^2=a^2-2ab+b^2}$

$\rm\implies \sqrt{\sqrt{5} -\sqrt{3}}^2$

$\rm\implies \sqrt{5}- \sqrt{3}$

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Now add the simplified forms of the terms:

$\rm\implies\sqrt{3} + \sqrt{2} +\sqrt{5} -\sqrt{3}$

$\rm\implies \sqrt{2} +\sqrt{5}$