# How do you simplify (5sqrt2+3sqrt5)(2sqrt10-5)?

Mar 24, 2017

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

#### Explanation:

You can multiply the brackets using the distributive law, in exactly the same way as in cases such as:

$\left(x + 3\right) \left(x - 4\right) = {x}^{2} - 4 x + 3 x - 12$

Each term in the first bracket must be multiplied by each term is the second bracket.

$\left(\textcolor{red}{5 \sqrt{2}} + \textcolor{b l u e}{3 \sqrt{5}}\right) \left(2 \sqrt{10} - 5\right)$

$= \textcolor{red}{5 \sqrt{2}} \left(2 \sqrt{10} - 5\right) + \textcolor{b l u e}{3 \sqrt{5}} \left(2 \sqrt{10} - 5\right)$

$= 10 \sqrt{20} - 25 \sqrt{2} + 6 \sqrt{50} - 15 \sqrt{5}$

Now find factors for the roots, using squares where possible:

=10sqrt(color(magenta)(4)xx5)-25sqrt2+6sqrt((color(lime)25xx2)-15sqrt5

$= 10 \times \textcolor{m a \ge n t a}{2} \sqrt{5} - 25 \sqrt{2} + 6 \times \textcolor{\lim e}{5} \sqrt{2} - 15 \sqrt{5}$

$= 20 \sqrt{5} - 15 \sqrt{5} + 30 \sqrt{2} - 25 \sqrt{2}$

$= 5 \sqrt{5} + 5 \sqrt{2}$

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

Mar 24, 2017

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

#### Explanation:

$\left(5 \sqrt{2} + 3 \sqrt{5}\right) \left(2 \sqrt{10} - 5\right)$

$\textcolor{w h i t e}{a a a a a a a a a a a a a}$$5 \sqrt{2} + 3 \sqrt{5}$
$\textcolor{w h i t e}{a a a a a a a a a a a a a}$$2 \sqrt{10} - 5$
$\textcolor{w h i t e}{a a a a a a a a a a a a a}$$- - - - -$
$\textcolor{w h i t e}{a a a a a a a a a a a a a}$$10 \sqrt{2} \sqrt{10} + 6 \sqrt{5} \sqrt{10}$
$\textcolor{w h i t e}{a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a}$$- 25 \sqrt{2} - 15 \sqrt{5}$
$\textcolor{w h i t e}{a a a a a a a a a a a a a}$$- - - - - - - - - - - - -$
$\textcolor{w h i t e}{a a a a a a a a a a a a}$color(blue)(10sqrt2sqrt10+6sqrt5sqrt10-25sqrt2-15sqrt5

$\therefore = 10 \sqrt{2} \times \sqrt{2} \times \sqrt{5} + 6 \sqrt{5} \times \sqrt{2} \times \sqrt{5} - 25 \sqrt{2} - 15 \sqrt{5}$

$\therefore = 10 \times 2 \sqrt{5} + 6 \times 5 \times \sqrt{2} - 25 \sqrt{2} - 15 \sqrt{5}$

$\therefore = 20 \sqrt{5} + 30 \sqrt{2} - 25 \sqrt{2} - 15 \sqrt{5}$

$\therefore = 20 \sqrt{5} - 15 \sqrt{5} + 30 \sqrt{2} - 25 \sqrt{2}$

$\therefore = 5 \sqrt{5} + 5 \sqrt{2}$

:.color(blue)(=5(sqrt2+sqrt5)