Question

square root of 4+root 15​

Answer 1

Answer:The result can be shown in multiple forms.

Exact Form:

4

√

4

15

Decimal Form:

15.49193338

Answer 2

Answer:

The value of the given root is

$\sqrt{4 + \sqrt{15} } = \frac{ \sqrt{3} + \sqrt{5} }{ \sqrt{2} }$

Step-by-step explanation:

The required square root which is need to be found us

$\sqrt{4 + \sqrt{15} }$

The above number is in form of a surd and we have a different methodology to find its root.We shall write this root as sum of two different arbitrary roots and later square on both sides after which we establish an relation where we finally get the value of the square root.

Let us assume that

$\sqrt{4 + \sqrt{15} } = \sqrt{a} + \sqrt{b}$

Squaring on both sides,we get

$4 + \sqrt{15} = a + b + 2 \sqrt{ab}$

From the above relation,we can establish two equations which are

$a + b = 4 - - > (1) \\ ab = \frac{15}{4} - - > (2)$

Using an relation,

$(a - b) {}^{2} = (a + b) {}^{2} - 4ab$

We get a different relation as follows

$(a - b) {}^{2} = {4}^{2} - 4 \times \frac{15}{4} = 1 \\ = > a - b = ±1 - - > (3)$

Substituting value of b from equation 2 in equation 1 we get,

$a + \frac{15}{4a} = 4 \\ = > 4 {a}^{2} + 15 = 16a \\ = > 4 {a}^{2} - 16a + 15 = 0 \\ = > a = \frac{3}{2} \: and \: \frac{5}{2}$

Therefore,

$a = \frac{3}{2} \: and \: b = \frac{5}{2}$

Even vice versa is possible but answer will be same.Hence our surd becomes

$\sqrt{a} + \sqrt{b} = \sqrt{ \frac{3}{2} } + \sqrt{ \frac{5}{2} } \\ = \frac{ \sqrt{3} + \sqrt{5} }{ \sqrt{2} }$

Therefore,the value of the root of the given number is

$\sqrt{4 + \sqrt{15} } = \frac{ \sqrt{3} + \sqrt{5} }{ \sqrt{2} }$

#SPJ2