How to Solve Rationalizing Imaginary Denominators? (+FREE Worksheet!)
How to solve rationalizing imaginary denominators, step by step: the rule in plain language, solved examples worked one line at a time, and practice questions so you can try it yourself before moving on.
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 to 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 to 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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