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Multiplying Complex Numbers

Multiplying complex numbers is very much like multiplying polynomials. When we simplify the result, we replace each occurrence of i2 with -1. We often write the final result in the form a + bi.

 

Example 1

Find: 5i · 7i

Solution

Multiply 5 · 7 and multiply i · i.

Replace i2 with -1.

Multiply.

So, 5i · 7i = -35.

5i · 7i

= 35 · i2

= 35 · (-1)

= -35

Note:

We write -35 in the form a + bi like this: 0 + (-35)i

 

Example 2

Find: 4i(9 - 6i)

Solution

Distribute 4i.

Multiply the factors in each term.

Replace i2 with -1.

Simplify.

Write the result in the form a + bi.

So, 4i(9 - 6i) = 24 + 36i.

  4i(9 - 6i)

= 4i · 9 - 4i · 6i

= 36i - 24i2

= 36i - 24(-1)

= 36i + 24

= 24 + 36i

 

Example 3

Find: (7 - 4i)(10 + 5i)

Solution

Multiply using the FOIL method.

Multiply the factors in each term.

Replace i2 with -1.

Combine like terms.

So, (7 - 4i)(10 + 5i) = 90 - 5i.

   (7 - 4i)(10 + 5i)

= 7 · 10 + 7 · 5i - 4i · 10 - 4i · 5i

= 70 + 35i - 40i - 20i2

= 70 + 35i - 40i - 20(-1)

= 70 + 35i - 40i + 20

= 90 - 5i

 
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