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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 Â· 7i

 Solution Multiply 5 Â· 7 and multiply i Â· i. Replace i2 with -1. Multiply. So, 5i Â· 7i = -35. 5i Â· 7i = 35 Â· i2 = 35 Â· (-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 Â· 9 - 4i Â· 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 Â· 10 + 7 Â· 5i - 4i Â· 10 - 4i Â· 5i = 70 + 35i - 40i - 20i2 = 70 + 35i - 40i - 20(-1) = 70 + 35i - 40i + 20 = 90 - 5i

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