How do you rationalize imaginary denominators?

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How do you rationalize imaginary denominators?

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Answer 1 · 2015-03-30T13:04:55

If $a + b i$ $a + bi$ is an imaginary number
its conjugate is $a - b i$ $a-bi$ (also an imaginary number)
and
the product of an imaginary number and its conjugate it not an imaginary number.
$(a + b i) \times (a - b i) = a^{2} - b^{2}$ $(a+bi)xx(a-bi) = a^2 - b^2$

If you have a number with an imaginary denominator multiply both the numerator and denominator by the conjugate of the denominator.

For example, suppose you want to rationalize the denominator of
$\frac{10}{3 + 2 i}$ $(10)/(3+2i)$

$\frac{10}{3 + 2 i} \cdot \frac{3 - 2 i}{3 - 2 i}$ $(10)/(3+2i) *(3-2i)/(3-2i)$

$= \frac{10 (3 - 2 i)}{3^{2} - 2^{2}}$ $=(10(3-2i))/(3^2 - 2^2)$

$= \frac{30 - 20 i}{9 - 4}$ $= (30-20i)/(9-4)$

$= 6 - 4 i$ $= 6-4i$

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How do you rationalize imaginary denominators?

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