Write the expression (square root of 12) * (square root of -4) / (square root of 3) in the form a+bi, where a and b are real numbers.
Thanks for any help!
$\displaystyle \frac{\sqrt{12}\times \sqrt{-4}}{\sqrt{3}} $
$\displaystyle = \frac{\sqrt{12}\times \sqrt{-1 \times 4}}{\sqrt{3}} $
$\displaystyle = \frac{\sqrt{12} \times \sqrt{-1} \times \sqrt{4}}{\sqrt{3}} $
$\displaystyle = \frac{\sqrt{12} \times i \times 2}{\sqrt{3}} $
$\displaystyle = 2i \times \frac{\sqrt{12}}{\sqrt{3}} $
$\displaystyle = 2i \times \sqrt{\frac{12}{3}} $
$\displaystyle = 2i \times \sqrt{4} $
$\displaystyle = 2i \times 2 $
$\displaystyle = 4i $
$\displaystyle = 0+4i $
If 12 was negative then you would do the exact same thing as I did with the -4.
$\displaystyle \sqrt{-12} = \sqrt{12 \times -1} $
$\displaystyle = \sqrt{12} \times \sqrt{-1} $
$\displaystyle = \sqrt{12} \times i $
$\displaystyle = i\sqrt{12} $
In general:
$\displaystyle \sqrt{-a} = i\sqrt{a} $.