How do you simplify (square root 5 + 9 square root 2)(4 square root 5 - square root 2)?

Jul 27, 2015

=color(blue)(2 +35sqrt(10)

Explanation:

Expressing the question in numbers:

$= \textcolor{b l u e}{\left(\sqrt{5} + 9 \sqrt{2}\right)} \cdot \left(4 \sqrt{5} - \sqrt{2}\right)$

Each term within the first bracket needs to be multiplied with each term within the second bracket.

$= \textcolor{b l u e}{\left(\sqrt{5}\right)} \cdot \left(4 \sqrt{5} - \sqrt{2}\right) + \textcolor{b l u e}{\left(9 \sqrt{2}\right)} \cdot \left(4 \sqrt{5} - \sqrt{2}\right)$

$= \left[\left(\sqrt{5}\right) \left(4 \sqrt{5}\right) + \left(\sqrt{5}\right) \left(- \sqrt{2}\right)\right] + \left[9 \left(\sqrt{2}\right) \left(4 \sqrt{5}\right) + \left(9 \sqrt{2}\right) \cdot \left(- \sqrt{2}\right)\right]$

$= \left[4 \cdot 5 - \sqrt{2 \cdot 5}\right] + \left[36 \sqrt{2 \cdot 5} - 18\right]$

$= \left[20 - \sqrt{10}\right] + \left[36 \sqrt{10} - 18\right]$

Grouping like terms
$= \left(20 - 18\right) - \sqrt{10} + 36 \sqrt{10}$

$= 2 + 35 \sqrt{10}$