## How to Solve Rationalizing Imaginary Denominators

### Rationalizing Imaginary Denominators

"Rationalizing the denominator" is moving a root (like a square root, cube root, or imaginary number) from the bottom of a fraction to the top.
For something to be in its "simplest form," the denominator can't be irrational. "Rationalizing the Denominator" is the process of making the denominator rational.
Note: There's nothing wrong with an irrational denominator; it still works. But it's not the "simplest form," so you might lose points for it. And taking them out might help you figure out how to solve an equation, so you should learn how. In this article, you'll learn how to make rational the denominators that are irrational.

### How to get rid of imaginary denominators, step by step

• Step 1: Find the conjugate, between the two terms, it is the denominator with a different sign.
• Step 2: Use the conjugate to multiply the numerator and denominator.
• Step 3: If necessary, simplify.

Look at these examples to see what this means:

### Example1

Solve $$\frac{2i}{3i \ + \ 1}$$

Solution:
Take the conjugate of $$3i \ + \ 1$$: $$3i \ - \ 1$$
Multiply the numerator and denominator by the conjugate:
$$\frac{2i}{3i \ + \ 1} \times \frac{3i \ - \ 1}{3i \ - \ 1} \ = \ \frac{6i^2 \ - \ 2i}{9i^2 \ - \ 1} \ = \ \frac{-6 \ - \ 2i}{-9 \ - \ 1} \ = \ \frac{-6 \ - \ 2i}{-10}$$

### Example2

Solve $$\frac{i \ + \ 3}{5i \ + \ 4}$$

Solution:
Take the conjugate of $$5i \ + \ 4$$: $$5i \ - \ 4$$
Multiply the numerator and denominator by the conjugate:
$$\frac{i \ + \ 3}{5i \ + \ 4} \times \frac{5i \ - \ 4}{5i \ - \ 4} \ = \ \frac{5i^2 \ + \ 4i \ +\ 15i^2 \ + \ 20}{25i^2 \ - \ 16} \ = \ \frac{-5 \ + \ 4i \ - \ 15 \ + \ 20}{-25 \ - \ 16} \ = \ \frac{4i}{-41}$$

### Exercises for Rationalizing Imaginary Denominators

1) Simplify: $$\frac{2}{3i}$$

2) Simplify: $$\frac{4}{i \ + \ 1}$$

3) Simplify: $$\frac{6}{i \ - \ 1}$$

4) Simplify: $$\frac{6}{i}$$

5) Simplify: $$\frac{2}{3i \ + \ 1}$$

6) Simplify: $$\frac{1}{4i \ + \ 2}$$

7) Simplify: $$\frac{-3i}{-2 \ - \ 3i}$$

8) Simplify: $$\frac{2i \ + \ 5}{i \ - \ 6}$$

9) Simplify: $$\frac{3 \ - \ i}{-2 \ + \ 4i}$$

10) Simplify: $$\frac{8 \ - \ 3i}{-i}$$

1) Simplify: $$\frac{2}{3i}$$

$$\color{red}{\frac{2}{3i} \ = \ \frac{2}{3i} \times \frac{i}{i} \ = \ \frac{2i}{-3}}$$

2) Simplify: $$\frac{4}{i \ + \ 1}$$

$$\color{red}{\frac{4}{i \ + \ 1} \ = \ \frac{4}{i \ + \ 1} \times \frac{i \ - \ 1}{i \ - \ 1} \ = \ \frac{4i \ - \ 4}{-1 \ - \ 1} \ = \ \frac{4i \ - \ 4}{-2} \ = \ 2 \ - \ 2i}$$

3) Simplify: $$\frac{6}{i \ - \ 1}$$

$$\color{red}{\frac{6}{i \ - \ 1} \ = \ \frac{6}{i \ - \ 1} \times \frac{i \ + \ 1}{i \ + \ 1} \ = \ \frac{6i \ + \ 6}{-1 \ - \ 1} \ = \ \frac{6i \ + \ 6}{-2} \ = \ -3 \ - \ 3i}$$

4) Simplify: $$\frac{6}{i}$$

$$\color{red}{\frac{6}{i} \ = \ \frac{6}{i} \times \frac{i}{i} \ = \ \frac{6i}{-1} \ = \ -6i}$$

5) Simplify: $$\frac{2}{3i \ + \ 1}$$

$$\color{red}{\frac{2}{3i \ + \ 1} \ = \ \frac{2}{3i \ + \ 1} \times \frac{3i \ - \ 1}{3i \ - \ 1} \ = \ \frac{6i \ - \ 2}{-9 \ - \ 1} \ = \ -\frac{6i}{10} \ + \ \frac{2}{10} \ = \ -\frac{3i}{5} \ + \ \frac{1}{5}}$$

6) Simplify: $$\frac{1}{4i \ + \ 2}$$

$$\color{red}{\frac{1}{4i \ + \ 2} \ = \ \frac{1}{4i \ + \ 2} \times \frac{4i \ - \ 2}{4i \ - \ 2} \ = \ \frac{4i \ - \ 2}{-16 \ - \ 4} \ = \ \frac{4i \ - \ 2}{-20} \ = \ -\frac{4i}{20} \ + \ \frac{2}{20} \ = \ -\frac{i}{5} \ + \ \frac{1}{10}}$$

7) Simplify: $$\frac{-3i}{-2 \ - \ 3i}$$

$$\color{red}{\frac{-3i}{-2 \ - \ 3i} \ = \ \frac{-3i}{-2 \ - \ 3i} \times \frac{-2 \ + \ 3i}{-2 \ + \ 3i} \ = \ \frac{6i \ + \ 9}{4 \ + \ 9} \ = \ \frac{6i \ + \ 9}{13} \ = \ \frac{6i}{13} \ + \ \frac{9}{13}}$$

8) Simplify: $$\frac{2i \ + \ 5}{i \ - \ 6}$$

$$\color{red}{\frac{2i \ + \ 5}{i \ - \ 6} \ = \ \frac{2i \ + \ 5}{i \ - \ 6} \times \frac{i \ + \ 6}{i \ + \ 6} \ = \ \frac{-2 \ + \ 12i \ + \ 5i \ + \ 30}{-1 \ - \ 36} \ = \ \frac{28 \ + \ 17i}{-37} \ = \ -\frac{28}{37} \ - \ \frac{17i}{37}}$$

9) Simplify: $$\frac{3 \ - \ i}{-2 \ + \ 4i}$$

$$\color{red}{\frac{3 \ - \ i}{-2 \ + \ 4i} \ = \ \frac{3 \ - \ i}{-2 \ + \ 4i} \times \frac{-2 \ - \ 4i}{-2 \ - \ 4i} \ = \ \frac{-6 \ - \ 12i \ + \ 2i \ - \ 4}{4 \ - \ (-16)} \ = \ \frac{-10 \ - \ 10i}{20} \ = \ -\frac{10}{20} \ - \ \frac{10i}{20} \ = \ -\frac{1}{2} \ - \ \frac{i}{2}}$$

10) Simplify: $$\frac{8 \ - \ 3i}{-i}$$

$$\color{red}{\frac{8 \ - \ 3i}{-i} \ = \ \frac{8 \ - \ 3i}{-i} \times \frac{i}{i} \ = \ \frac{8i \ + \ 3}{-(-1)} \ = \ 8i \ + \ 3}$$

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