The following step-by-step guide helps you learn how to rationalize imaginary denominators.

**Step by step guide to rationalizing Imaginary Denominators**

- Step 1: Find the conjugate (it’s the denominator with different sign between the two terms.
- Step 2: Multiply the numerator and denominator by the conjugate.
- Step 3: Simplify if needed.

### Example 1:

Solve: \(\frac{2-3i}{6i}\)

**Solution:**

Multiply by the conjugate: \(\frac{-i}{-i}\).

\(\frac{2-3i}{6i}=\frac{(2-3i)(-i)}{6i(-i) }=\frac{-3-2i}{6}=-\frac{1}{2}-\frac{1}{3} i\)

### Example 2:

Solve: \(\frac{8i}{2 – 4i}\)

**Solution**:

Factor \(2 – 4i=2(1-2i)\), then: \(\frac{8i}{2(1-2i)}=\frac{4i}{(1-2i)}\)

Multiply by the conjugate \(\frac{1+2i}{1+2i}: \frac{4i(1+2i)}{(1-2i)(1+2i)}=\frac{-8+4i}{5}=-\frac{8}{5}+\frac{4}{5} i\)

### Example 3:

Solve: \(\frac{5i}{2 – 3i}\)

**Solution**:

Multiply by the conjugate: \(\frac{2+ 3i}{2+ 3i}→\frac{5i(2+ 3i)}{(2-3i)(2+ 3i)}=\frac{-15+10i}{(2-3i)(2+ 3i)}\)

Use complex arithmetic rule: \((a+bi)(a-bi)=a^2+b^2\)

\( (2-3i)(2+ 3i)=-2^2+(-3)^2=13\) , Then: \(\frac{-15+10i}{(2-3i)(2+ 3i)}=\frac{-15+10i}{13}\)

### Example 4:

Solve: \(\frac{4-9i}{-6i}\)

**Solution**:

Apply fraction rule: \(\frac{4-9i}{-6i}=-\frac{4-9i}{6i}\)

Multiply by the conjugate: \(\frac{-i}{-i}\).

\( -\frac{4-9i}{6i}=-\frac{(4-9i)(-i)}{6i(-i)} =-\frac{-9-4i}{6}\)

## Exercises

### Simplify.

- \(\color{blue}{\frac{10 – 10i}{- 5i}} \\\ \)
- \(\color{blue}{\frac{4 – 9i}{- 6i}} \\\ \)
- \(\color{blue}{\frac{6 + 8i}{9i}} \\\ \)
- \(\color{blue}{\frac{8i}{-1+3i}} \\\ \)
- \(\color{blue}{\frac{5i}{- 2 – 6i}} \\\ \)
- \(\color{blue}{\frac{- 10 – 5i}{- 6 + 6i}} \\\ \)

### Download Rationalizing Imaginary Denominators Worksheet

- \(\color{blue}{2i + 2} \\\ \)
- \(\color{blue}{\frac{4 + 9i}{6}} \\\ \)
- \(\color{blue}{\frac{6+8i}{9}} \\\ \)
- \(\color{blue}{\frac{-4i+12}{5}} \\\ \)
- \(\color{blue}{\frac{-i-3}{4}} \\\ \)
- \(\color{blue}{\frac{5+15i}{12}} \\\ \)