Question

evaluate
$\sqrt{5 + 2 \sqrt{6} } + \sqrt{8 - 2 \sqrt{15} }$
​

Answer 1

Question :-

• Evaluate √[ 5 + 2√6 ] + √[8 - 2√15]

Solution :-

→ √[ 5 + 2√6 ]

→ √[ 2 + 3 + 2 * √(2*3) ]

→ √[(√2)² + (√3)² + 2 * √2 * √3 ]

→ a² + b² + 2ab = (a + b)²

→ √[(√2 + √3)²]

→ √(a)² = a

→ (√2 + √3) .

__________________

Similarly,

→ √[8 - 2√15]

→ √[5 + 3 - 2 √(5*3) ]

→ √[(√5)² + (√3)² - 2 * √5 * √3]

→ a² + b² - 2ab = (a - b)²

→ √[(√5 - √3)²]

→ √(a)² = a

→ (√5 - √3) .

__________________

So,

→ √[ 5 + 2√6 ] + √[8 - 2√15]

→ (√2 + √3) + (√5 - √3)

→ (√2 + √5) (Ans).

Answer 2

We know that,

$( \sqrt{a} + \sqrt{b} ) {}^{2} = a + b + 2 \sqrt{ab}$

Now,

$\sqrt{5} + 2 \sqrt{6} = a + b + 2 \sqrt{a}$

$\star \: a + b = \sqrt{5} \rightarrow \: ab = 6 \rightarrow \: a = 3 \: and \: b = 2$

$5 + 2 \sqrt{6} = 3 + 2 + 2 \sqrt{} (3 \times 2)$

$5 + 2 \sqrt{6} = ( \sqrt{3} + \sqrt{2} ) {}^{2}$

$\therefore \: 5 + 2 \sqrt{6} = ( \sqrt{3} + \sqrt{2} )$

Similarly,

$\sqrt{8} - 2 \sqrt{15 } = a + b - 2 \sqrt{ab}$

$\star \: a + b = 8 \rightarrow \: ab = 15 \rightarrow \: a = 5 \: and \: b = 3$

$\sqrt{8} - 2 \sqrt{15} = 5 + 3 - 2 \sqrt{} (5 \times 3)$

$\sqrt{8} - 2 \sqrt{15 } = ( \sqrt{5} - \sqrt{3} ) {}^{2}$

$\sqrt{8} - 2 \sqrt{15} = ( \sqrt{5} - \sqrt{3} )$

According to the question:

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

$= \sqrt{3} + \sqrt{2} + \sqrt{5} - \sqrt{3}$

$= \sqrt{2} + \sqrt{5}$