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!

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- Jan 18th 2009, 12:51 PMlive_laugh_luv27Rewriting 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 PMMush
$\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 PMlive_laugh_luv27
Thanks!

What if 12 was negative, how would the problem change??

Thanks again for your help. - Jan 18th 2009, 01:31 PMMush
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} $.