How to Solve Rationalizing Imaginary Denominators? (+FREE Worksheet!)
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: \(\frac{2-3i}{6i}\)
Solve Rationalizing Imaginary Denominators: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Separate partsKeep real and imaginary terms in their own lanes.
- Use i squaredReplace \(i^2\) with -1 whenever it appears.
- Use conjugatesFor division, multiply by the conjugate to make the denominator real.
Worked examples
Add complex numbers
- Add real parts: 4 + 2.
- Add imaginary parts: 3i – 5i.
- Write both parts together.
Use i squared
- Multiply coefficients to get 5.
- i times i is i squared.
- Replace i squared with -1.
Try one before moving on
Solve Rationalizing Imaginary Denominators: pop-up practice
Solution:
Multiply by the conjugate: \(\frac{-i}{-i}\):
\(\frac{2-3i}{6i}=\frac{(2-3i)(-i)}{6i(-i) }=\frac{-3-2i}{6}=-\frac{1}{2}-\frac{1}{3} i\)
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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 \(\frac{1+2i}{1+2i}\):
\(\frac{4i}{1- 2i}= \frac{4i(1+2i)}{(1-2i)(1+2i)}=\frac{-8+4i}{5}=-\frac{8}{5}+\frac{4}{5} i\)
Rationalizing Imaginary Denominators – Example 3:
Solve: \(\frac{5i}{2 – 3i}\)
Solution:
Multiply by the conjugate: \(\frac{2+ 3i}{2+ 3i}\):
\(\frac{5i}{2 – 3i}=\frac{5i(2+ 3i)}{(2-3i)(2+ 3i)}=\frac{-15+10i}{(2-3i)(2+ 3i)}\)
Use complex arithmetic rule: \((a+bi)(a-bi)=a^2+b^2\)
\( (2-3i)(2+ 3i)=-2^2+(-3)^2=4+9=13\),
Then: \(\frac{-15+10i}{(2-3i)(2+ 3i)}=\frac{-15+10i}{13}= \frac {-15}{13}+ \frac{10}{13} i\)
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: \(\frac{-i}{-i}\).
\( -\frac{4-9i}{6i}=-\frac{(4-9i)(-i)}{6i(-i)} =-\frac{-9-4i}{6}\)\(=\frac {3}{2} + \frac{2}{3}i\)
Exercises for Solving Rationalizing Imaginary Denominators
Simplify.
- \(\color{blue}{\frac{10 – 10i}{- 5i}} \\\ \)
- \(\color{blue}{\frac{5 – 8i}{- 10i}} \\\ \)
- \(\color{blue}{\frac{6 + 8i}{9i}} \\\ \)
- \(\color{blue}{\frac{8i}{-1+3i}} \\\ \)
- \(\color{blue}{\frac{5i}{- 2 – 6i}} \\\ \)
- \(\color{blue}{\frac{- 10 – 5i}{- 6 + 6i}} \\\ \)
Download Rationalizing Imaginary Denominators Worksheet
- \(\color{blue}{2+ 2i} \\\ \)
- \(\color{blue}{\frac{4}{5}+\frac {1}{2}i} \\\ \)
- \(\color{blue}{\frac{8}{9}-\frac{2}{3}i} \\\ \)
- \(\color{blue}{\frac{12}{5}-\frac{4}{5}i}\\\ \)
- \(\color{blue}{\frac{-3}{4}-\frac{1}{4}i} \\\ \)
- \(\color{blue}{\frac{5}{12}+\frac{5}{4}i} \\\ \)
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