Question

what is the value of
$\sqrt{15 + 2 \sqrt{10} }$
? ​

Answer 1

Step-by-step explanation:

Let's take the √(15+2√10) be √x +√y.

On squaring on both sides we get

${( \sqrt{x} + \sqrt{y})}^{2} = 15 + 2 \sqrt{10} = >$

$x + y + 2 \sqrt{xy} = 15 + 2 \sqrt{10} = >$

So

x+y=15 and xy=10 =>x= 10/y hence

$y + \frac{10}{y} = 15 = > {y}^{2} + 10 = 15y = >$

${y}^2 - 15y + 10 = 0 = >$

$y = \frac{15 + \sqrt{ {15}^{2} - 4 \times 10} }{2} \: or \: \frac{15 - \sqrt{ {15}^{2} - 4 \times 10 } }{2}$

=> y= (15+√185)/2 or (15-√185)/2. So

x+y = 15 => x = 15-(15+√185)/2 or 15-(15-√185)/2 = (15+√185)/2 or

(15-√185)/2 So the square root of 15+2√10

$= > \sqrt{ \frac{15 + \sqrt{185} }{2} } + \sqrt{ \frac{15 - \sqrt{185} }{2} } = >$

=> 3.7816313 + 0.836220512 =>

4.6178581 =>4.618 approximately. Hope it helps you.