Question

$\sqrt{11 - 4 \sqrt{7} }$
please find the value of the above question​

Answer 1

$\huge{\bold{SOLUTION:-}}$

$\sqrt{11 - 4 \sqrt{7} } = \sqrt{x} - \sqrt{y}$

Squaring on both sides

$( { \sqrt{11 - 4 \sqrt{7} } })^{2} = ( { \sqrt{x} - \sqrt{y} })^{2}$

After equating the rational and irrational parts on both sides , we get

$x + y = 11$

$2 \sqrt{x} \sqrt{y} = 4 \sqrt{7}$

$4xy = 112$

$( {x - y)}^{2} = {(x + y)}^{2} - 4xy$

$121 - 112$

$\boxed{x - y = ± 3}$

Therefore ,

$\sqrt{11 - 4 \sqrt{7} } = \sqrt{7} - \sqrt{4} \: or \: \sqrt{4} - \sqrt{7}$

Must-to-remember :-

Surd is simply one form of an irrational number

if n is positive integer and a is rational number and is not the nth power of any rational number , then n root a is called a surd

Surd is a number that can't be simplified to remove a square root or cube root etc.

A surd is a square root which cannot be reduced to a whole number

When it is a root and irrational , it is a surd

But all roots are not surds

In the above given problem, the trick is just finding the simplified number from the root and adding or subtracting or multiplying or division etc..

Answer 2

$\sqrt{11 - 4 \sqrt{7} }$

Let

$\sqrt{11 - 4 \sqrt{7} } = \sqrt{x} - \sqrt{y}$

Then

$\sqrt{11 - 4 \sqrt{7} } = \sqrt{x} - \sqrt{y} \\ \\ 11 - 4 \sqrt{7} = ( \sqrt{x} - \sqrt{y) } {}^{2} \\ \\ 11 - 4 \sqrt{7} = x + y - 2 \sqrt{xy }$

Comparing Both Sides We get

$x + y = 11 \\ \\ - 2 \sqrt{xy} = - 4 \sqrt{7} \\ \\ \sqrt{xy} = \frac{ - 4 \sqrt{7} }{ - 2} \\ \\ \sqrt{xy} = 2 \sqrt{7} \\ \\ xy =( 2 \sqrt{7} ) {}^{2} \\ \\ x \times y = 4 \times 7$

We get

$x \times y = 4 \times 7 \\ \\ x + y =4 + 7 = 11$

Comparing Both Sides We get

$x = 7\\ \\ y = 4$

As we know

${11 - 4 \sqrt{7} } = (\sqrt{x} - \sqrt{y} ) {}^{2} \\ \\ 11 - 4 \sqrt{7} = ( \sqrt{7} - \sqrt{4} ) {}^{2} \\ \\ 11 - 4 \sqrt{7} = ( \sqrt{7} - 2) {}^{2}$

$\sqrt{11+4\sqrt{7}}=(\sqrt{7}-{2})$