# How do you perform the operation and write the result in standard form given (3+sqrt(-5))(7-sqrt(-10))?

May 7, 2017

$\left(3 + \sqrt{- 5}\right) \left(7 - \sqrt{- 10}\right) = \left(21 + 5 \sqrt{2}\right) - \left(3 \sqrt{10} + 7 \sqrt{5}\right) i$

#### Explanation:

$\left(3 + \sqrt{- 5}\right) \left(7 - \sqrt{- 10}\right)$

= 3×7-3×sqrt(-10)-7sqrt(-5)+sqrt(-5)sqrt(-10)

= 21-3sqrt((-1)×10)-7sqrt((-1)×5)+sqrt((-5)×(-10))

= 21-3sqrt10×sqrt(-1)-7sqrt5×sqrt(-1)+sqrt50

= $\left(21 + \sqrt{50}\right) - \left(3 \sqrt{10} + 7 \sqrt{5}\right) \sqrt{- 1}$

= $\left(21 + 5 \sqrt{2}\right) - \left(3 \sqrt{10} + 7 \sqrt{5}\right) i$

as $\sqrt{- 1} = i$