How to Solve Rationalizing Imaginary Denominators

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.

Rationalizing Imaginary Denominators – Example 1:

Solve: $$\frac{2-3i}{6i}$$

Solution:

Multiply by the conjugate: $$\frac{-i}{-i}$$.
$$\frac{2-3i}{6i}=\frac{(2-3i)(-i)}{6i(-i) }=\frac{-3-2i}{6}=-\frac{1}{2}-\frac{1}{3} i$$

Rationalizing Imaginary Denominators – 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 $$\frac{1+2i}{1+2i}: \frac{4i(1+2i)}{(1-2i)(1+2i)}=\frac{-8+4i}{5}=-\frac{8}{5}+\frac{4}{5} i$$

Rationalizing Imaginary Denominators – 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: $$\frac{-15+10i}{(2-3i)(2+ 3i)}=\frac{-15+10i}{13}$$

Rationalizing Imaginary Denominators – 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 for Solving Rationalizing Imaginary Denominators

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}} \\\$$

• $$\color{blue}{2i + 2} \\\$$
• $$\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}} \\\$$

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