# How do you simplify (3+ sqrt 5) times (5- sqrt5)?

Jun 4, 2016

$10 + 2 \sqrt{5}$

#### Explanation:

By expanding the brackets using FOIL or similar method.

$\Rightarrow \left(3 + \sqrt{5}\right) \left(5 - \sqrt{5}\right) = 15 - 3 \sqrt{5} + 5 \sqrt{5} - {\left(\sqrt{5}\right)}^{2}$
$\text{---------------------------------------------------------------}$

now in general $\sqrt{a} \times \sqrt{a} = a$

$\Rightarrow {\left(\sqrt{5}\right)}^{2} = \sqrt{5} \times \sqrt{5} = 5$

radicals may be collected together in a similar manner to 'collecting like terms' algebraically.

That is $4 \sqrt{3} + 5 \sqrt{3} = \left(4 + 5\right) \sqrt{3} = 9 \sqrt{3}$

$\Rightarrow - 3 \sqrt{5} + 5 \sqrt{5} = \left(- 3 + 5\right) \sqrt{5} = 2 \sqrt{5}$
$\text{---------------------------------------------------------------}$

$\Rightarrow \left(3 + \sqrt{5}\right) \left(5 - \sqrt{5}\right) = 15 - 3 \sqrt{5} + 5 \sqrt{5} - {\left(\sqrt{5}\right)}^{2}$

$= 15 + 2 \sqrt{5} - 5 = 10 + 2 \sqrt{5}$