Question

$\sqrt{8 + 2 \sqrt{15} } - \sqrt{8 - 2 \sqrt{15} }$
is:
(a)
$2 \sqrt{15}$
(b)
$\sqrt{8}$
(c)
$\sqrt{12}$
(d) √5​

Answer 1

Answer:

√12

Step-by-step explanation:

Let √(8 + 2√15) - √(8 - 2√15) be x.

Square on both sides :

=> [√(8 + 2√15) - √(8 - 2√15)]² = x²

=> √(8 + 2√15)² + √(8 - 2√15)² - 2√(8 + 2√15)(8 - 2√15) = x²

=> (8 + 2√15) + (8 - 2√15) - 2√(8² - (2√15)²) = x²

=> 16 - 2√(64 - 60) = x²

=> 16 - 2(2) = x²

=> 12 = x²

=> √12 = x

As we assumed '√(8 + 2√15) - √(8 - 2√15) be x',

√(8 + 2√15) - √(8 - 2√15) is √12

This can also be solved as :

=> √(8 + 2√15) - √(8 - 2√15)

=> √(5 + 3 + 2√5√3) - √(5 + 3 - 2√5√3)

=> √(√5² + √3² + 2√5√3) - √(√5² + √3² - 2√5√3)

=> √(√5 + √3)² - √(√5 - √3)²

=> (√5 + √3) - (√5 - √3)

=> 2√3

=> √3*4

=> √12

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Answer 2

Answer:

Solution :-

Rationalising

$\sf \sqrt{8 + 2 \sqrt{15} } - \sqrt{8 - 2 \sqrt{15} }$

8 can be written as 3 + 5 or 5 + 3

$\sf \sqrt{5 + 3 + 2 \sqrt{15} } - \sqrt{5 + 3 - 2 \sqrt{15} }$

$\sf \: \sqrt{5 + 3 + 2 \sqrt{5} \times \sqrt{3} } - \sqrt{5 + 3 - 2 \sqrt{5} \times \sqrt{3} }$

$\sf \: \sqrt{\sqrt{5} {}^{2} + \sqrt{3 }{}^{2} } - \sqrt{ \sqrt{5} {}^{2} - \sqrt{3 {}^{2} } }$

$\sf \: \sqrt{5} + \sqrt{3} - \sqrt{5} + \sqrt{3}$

$\sf \: \sqrt{3} + \sqrt{3}$

$\sf \: 2 \sqrt{3}$

$\sf \: \sqrt{12}$