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Presentation transcript:

You can multiply complex numbers by using the Distributive Property and treating the imaginary parts as like terms. Simplify by using the fact i2 = –1.

Example 5A: Multiplying Complex Numbers Multiply. Write the result in the form a + bi. –2i(2 – 4i) Distribute. –4i + 8i2 –4i + 8(–1) Use i2 = –1. –8 – 4i Write in a + bi form.

Example 5B: Multiplying Complex Numbers Multiply. Write the result in the form a + bi. (3 + 6i)(4 – i) 12 + 24i – 3i – 6i2 Multiply. 12 + 21i – 6(–1) Use i2 = –1. 18 + 21i Write in a + bi form.

Example 5C: Multiplying Complex Numbers Multiply. Write the result in the form a + bi. (2 + 9i)(2 – 9i) Multiply. 4 – 18i + 18i – 81i2 4 – 81(–1) Use i2 = –1. 85 Write in a + bi form.

Example 5D: Multiplying Complex Numbers Multiply. Write the result in the form a + bi. (–5i)(6i) –30i2 Multiply. –30(–1) Use i2 = –1 30 Write in a + bi form.

Multiply. Write the result in the form a + bi. Check It Out! Example 5a Multiply. Write the result in the form a + bi. 2i(3 – 5i) 6i – 10i2 Distribute. 6i – 10(–1) Use i2 = –1. 10 + 6i Write in a + bi form.

Multiply. Write the result in the form a + bi. Check It Out! Example 5b Multiply. Write the result in the form a + bi. (4 – 4i)(6 – i) 24 – 4i – 24i + 4i2 Distribute. 24 – 28i + 4(–1) Use i2 = –1. 20 – 28i Write in a + bi form.

Multiply. Write the result in the form a + bi. Check It Out! Example 5c Multiply. Write the result in the form a + bi. (3 + 2i)(3 – 2i) 9 + 6i – 6i – 4i2 Distribute. 9 – 4(–1) Use i2 = –1. 13 Write in a + bi form.

The imaginary unit i can be raised to higher powers as shown below. Notice the repeating pattern in each row of the table. The pattern allows you to express any power of i as one of four possible values: i, –1, –i, or 1. Helpful Hint

Example 6A: Evaluating Powers of i Simplify –6i14. –6i14 = –6(i2)7 Rewrite i14 as a power of i2. = –6(–1)7 = –6(–1) = 6 Simplify.

Example 6B: Evaluating Powers of i Simplify i63. Rewrite as a product of i and an even power of i. i63 = i  i62 = i  (i2)31 Rewrite i62 as a power of i2. = i  (–1)31 = i  –1 = –i Simplify.

Rewrite as a product of i and an even power of i. Check It Out! Example 6a Simplify . Rewrite as a product of i and an even power of i. Rewrite i6 as a power of i2. Simplify.

Simplify i42. i42 = ( i2)21 Rewrite i42 as a power of i2. Simplify. Check It Out! Example 6b Simplify i42. i42 = ( i2)21 Rewrite i42 as a power of i2. = (–1)21 = –1 Simplify.