# Rewriting imaginary number expression

• Jan 18th 2009, 12:51 PM
live_laugh_luv27
Rewriting imaginary number expression
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!
• Jan 18th 2009, 12:55 PM
Mush
Quote:

Originally Posted by live_laugh_luv27
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!

$\displaystyle \frac{\sqrt{12}\times \sqrt{-4}}{\sqrt{3}}$

$\displaystyle = \frac{\sqrt{12}\times \sqrt{-1 \times 4}}{\sqrt{3}}$

$\displaystyle = \frac{\sqrt{12} \times \sqrt{-1} \times \sqrt{4}}{\sqrt{3}}$

$\displaystyle = \frac{\sqrt{12} \times i \times 2}{\sqrt{3}}$

$\displaystyle = 2i \times \frac{\sqrt{12}}{\sqrt{3}}$

$\displaystyle = 2i \times \sqrt{\frac{12}{3}}$

$\displaystyle = 2i \times \sqrt{4}$

$\displaystyle = 2i \times 2$

$\displaystyle = 4i$

$\displaystyle = 0+4i$
• Jan 18th 2009, 01:07 PM
live_laugh_luv27
Thanks!
What if 12 was negative, how would the problem change??
• Jan 18th 2009, 01:31 PM
Mush
Quote:

Originally Posted by live_laugh_luv27
Thanks!
What if 12 was negative, how would the problem change??

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:

$\displaystyle \sqrt{-a} = i\sqrt{a}$.