How to Solve Rationalizing Imaginary Denominators

How to Solve Rationalizing Imaginary Denominators

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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 2i3i + 1

Solution:
Take the conjugate of 3i + 1: 3i  1
Multiply the numerator and denominator by the conjugate:
2i3i + 1×3i  13i  1 = 6i2  2i9i2  1 = 6  2i9  1 = 6  2i10

Example2

Solve i + 35i + 4

Solution:
Take the conjugate of 5i + 4: 5i  4
Multiply the numerator and denominator by the conjugate:
i + 35i + 4×5i  45i  4 = 5i2 + 4i + 15i2 + 2025i2  16 = 5 + 4i  15 + 2025  16 = 4i41

Free printable Worksheets

Exercises for Rationalizing Imaginary Denominators

1) Simplify: 23i

2) Simplify: 4i + 1

3) Simplify: 6i  1

4) Simplify: 6i

5) Simplify: 23i + 1

6) Simplify: 14i + 2

7) Simplify: 3i2  3i

8) Simplify: 2i + 5i  6

9) Simplify: 3  i2 + 4i

10) Simplify: 8  3ii

 

1) Simplify: 23i

23i = 23i×ii = 2i3

2) Simplify: 4i + 1

4i + 1 = 4i + 1×i  1i  1 = 4i  41  1 = 4i  42 = 2  2i

3) Simplify: 6i  1

6i  1 = 6i  1×i + 1i + 1 = 6i + 61  1 = 6i + 62 = 3  3i

4) Simplify: 6i

6i = 6i×ii = 6i1 = 6i

5) Simplify: 23i + 1

23i + 1 = 23i + 1×3i  13i  1 = 6i  29  1 = 6i10 + 210 = 3i5 + 15

6) Simplify: 14i + 2

14i + 2 = 14i + 2×4i  24i  2 = 4i  216  4 = 4i  220 = 4i20 + 220 = i5 + 110

7) Simplify: 3i2  3i

3i2  3i = 3i2  3i×2 + 3i2 + 3i = 6i + 94 + 9 = 6i + 913 = 6i13 + 913

8) Simplify: 2i + 5i  6

2i + 5i  6 = 2i + 5i  6×i + 6i + 6 = 2 + 12i + 5i + 301  36 = 28 + 17i37 = 2837  17i37

9) Simplify: 3  i2 + 4i

3  i2 + 4i = 3  i2 + 4i×2  4i2  4i = 6  12i + 2i  44  (16) = 10  10i20 = 1020  10i20 = 12  i2

10) Simplify: 8  3ii

8  3ii = 8  3ii×ii = 8i + 3(1) = 8i + 3

Rationalizing Imaginary Denominators Practice Quiz