When you have an exponent of i, check to see if it 4 or less. If so, simplify. If not, find as many powers of 4 as you can, because each i^{4} equals 1, which will multiply out without consequence.

0 + 3i + 8i + 21(-1) -21 + 11i

Remember to put the real number portion first and the imaginary part second.

Example 2

Simplify.

(4i + 3)(-2i + 1)(1 – i)^{2}

Apply FOIL as needed.

(-8i^{2} + 4i – 6i + 3)(1 – 2i + i^{2})

Simplifying after each multiplication can greatly reduce the complexity of what you are working with, because each term will reduce to either a real or imaginary component.

(-8(-1) – 2i + 3)(1 – 2i + (-1))

(11 – 2i)(-2i)

-22i + 4i^{2}

-4 – 22i

Example 3

Simplify.

5i^{6} + 3 + (2i^{2} – 1)(3i – 7)

5i^{2}(i^{4}) + 3 + (2(-1) – 1)(3i – 7)

5(-1)(1) + 3 + (-2 – 1)(3i – 7)

-2 – 3(3i – 7)

You can start off by simplifying some of the imaginary terms.

-2 – 9i + 21

19 – 9i

Equations with complex numbers are often not as complex as they first appear.