Explore: Rationalizing Denominators with Complex Numbers Solutions

Rationalizing Denominators with Complex Numbers Solutions

A number cannot be rational ℚ and imaginary i at the same time.
In any expression with complex numbers ℂ , the denominator must be rationalized so that it does not contain the imaginary unit.
The imaginary unit is not part of the set of real numbers; therefore it is not a rational number .
When the imaginary unit is in the denominator , apply the rules for rationalizing expressions and using conjugates so that the denominator only contains rational numbers.

Example 5

Simplify.

$\frac{3}{\sqrt{- 32}}$

Implement

$\frac{3}{i \sqrt{2^{5}}} = \frac{3}{i (2^{\frac{5}{2}})} = \frac{3}{4 i \sqrt{2}}$

$\frac{3}{4 i \sqrt{2}} (\frac{i \sqrt{2}}{i \sqrt{2}}) \frac{3 i \sqrt{2}}{4 i^{2} (2)} = \frac{3 i \sqrt{2}}{4 (- 1) (2)} - \frac{3 i \sqrt{2}}{8}$

Explain

Simplify the denominator
Rationalize

Note

The number 4 in the denominator is a rational coefficient. It is not necessary to include this value when rationalizing because it will simplify out.

Simplify

Example 6

Simplify.

$\frac{8}{5 - 3 i}$

Implement

$\frac{8}{5 - 3 i} (\frac{5 + 3 i}{5 + 3 i}) \frac{40 + 24 i}{25 - 9 i^{2}} = \frac{40 + 24 i}{25 + 9} \frac{40 + 24 i}{34} = \frac{2 (20 + 12 i)}{2 (17)} \frac{20 + 12 i}{17}$

Explain

Multiply by the conjugate and simplify