Fallbasiss4

2022-01-30

How do you simplify $\left(7\sqrt{-3}\right)\left(2\sqrt{-27}\right)?$

Amina Hall

Step 1
If you ignore the rule of imaginary numbers which states that
$\sqrt{-1}=i$ and ${i}^{2}=-1$
you would be tempted to multiply all terms together to get
$14\sqrt{81}=14×9=126$
which would be incorrect.
To keep the validity of the complex number system, always deal with the imaginary part first. To do this, rewrite the equation:
$\left(7\sqrt{3}×\sqrt{-1}\right)\left(2\sqrt{27}×\sqrt{-1}\right)=\left(\sqrt{-1}×\sqrt{-1}\right)\left(7\sqrt{3}×2\sqrt{27}\right)$
Now you can multiply terms within the parentheses.
$\left(\sqrt{-1}×\sqrt{-1}\right)=i×i={i}^{2}=-1$
and $\left(7\sqrt{3}×2\sqrt{27}\right)=14\sqrt{81}=14×9=126$
Multiply these together and you get $126×-1=-126$

Do you have a similar question?