# How do you simplify (sqrt(15)+sqrt(35)+sqrt(21)+5)/(sqrt(3)+2sqrt(5)+sqrt(7)) to get (sqrt(7)+sqrt(3))/2 ?

Sep 1, 2017

See explanation...

#### Explanation:

Given:

$\frac{\sqrt{15} + \sqrt{35} + \sqrt{21} + 5}{\sqrt{3} + 2 \sqrt{5} + \sqrt{7}}$

We could rationalise the denominator by multiplying both numerator and denominator by:

$\left(\sqrt{3} + 2 \sqrt{5} - \sqrt{7}\right) \left(\sqrt{3} - 2 \sqrt{5} + \sqrt{7}\right) \left(\sqrt{3} - 2 \sqrt{5} - \sqrt{7}\right)$

...but since we have been given what is at least supposed to be the answer, we can work backwards instead...

Note that:

$\left(\sqrt{7} + \sqrt{3}\right) \left(\sqrt{3} + 2 \sqrt{5} + \sqrt{7}\right)$

$= \sqrt{7} \left(\sqrt{3} + 2 \sqrt{5} + \sqrt{7}\right) + \sqrt{3} \left(\sqrt{3} + 2 \sqrt{5} + \sqrt{7}\right)$

$= \sqrt{21} + 2 \sqrt{35} + 7 + 3 + 2 \sqrt{15} + \sqrt{21}$

$= 2 \left(\sqrt{15} + \sqrt{35} + \sqrt{21} + 5\right)$

So, dividing both ends by $2 \left(\sqrt{3} + 2 \sqrt{5} + \sqrt{7}\right)$ we get:

$\frac{\sqrt{15} + \sqrt{35} + \sqrt{21} + 5}{\sqrt{3} + 2 \sqrt{5} + \sqrt{7}} = \frac{\sqrt{7} + \sqrt{3}}{2}$