Question

find the square root of
$11 + 2 \sqrt{30}$
Plz Tell me the right Answer ​

Answer 1

$\large\underline{\bold{Given\:Question-}}$

$\sf \: Find : \: \sqrt{11 + 2 \sqrt{30} }$

$\large\underline{\sf{Solution-}}$

Consider,

$\rm :\longmapsto\: \sqrt{11 + 2 \sqrt{30} }$

can be re-written as

$\rm \: \: = \: \: \sqrt{6 + 5 + 2 \sqrt{6 \times 5}}$

$\rm \: \: = \: \: \sqrt{ {( \sqrt{6})}^{2} + {( \sqrt{5} )}^{2} + 2 \sqrt{6} \times \sqrt{5}}$

We know,

$\boxed{ \sf \: {x}^{2} + {y}^{2} + 2xy = {(x + y)}^{2}}$

So, using this identity, the above step can be rewritten as

$\rm \: \: = \: \: \sqrt{ {\bigg( \sqrt{6} + \sqrt{5} \bigg) }^{2} }$

$\rm \: \: = \: \: \sqrt{6} + \sqrt{5}$

$\bf\implies \: \sqrt{11 + 2 \sqrt{30}} = \sqrt{6} + \sqrt{5}$

Additional Information :-

More Identities to know:

• (a + b)² = a² + 2ab + b²

• (a - b)² = a² - 2ab + b²

• a² - b² = (a + b)(a - b)

• (a + b)² = (a - b)² + 4ab

• (a - b)² = (a + b)² - 4ab

• (a + b)² + (a - b)² = 2(a² + b²)

• (a + b)³ = a³ + b³ + 3ab(a + b)

• (a - b)³ = a³ - b³ - 3ab(a - b)