Math Problem Statement

(2 - 6i)(4 - 2i)

Solution

The expression provided in the image is:

(26i)(42i)(2 - 6i)(4 - 2i)

Let's multiply these two complex numbers using the distributive property (FOIL method):

(26i)(42i)=24+2(2i)+(6i)4+(6i)(2i)(2 - 6i)(4 - 2i) = 2 \cdot 4 + 2 \cdot (-2i) + (-6i) \cdot 4 + (-6i) \cdot (-2i)

Now, calculate each term:

  1. 24=82 \cdot 4 = 8
  2. 2(2i)=4i2 \cdot (-2i) = -4i
  3. (6i)4=24i(-6i) \cdot 4 = -24i
  4. (6i)(2i)=12i2(-6i) \cdot (-2i) = 12i^2, and since i2=1i^2 = -1, this becomes 12(1)=1212(-1) = -12

Now, combine all terms:

84i24i12=(812)+(4i24i)=428i8 - 4i - 24i - 12 = (8 - 12) + (-4i - 24i) = -4 - 28i

So, the product is:

428i-4 - 28i

Would you like more details on this solution or have any questions?

Here are some related questions:

  1. How do you multiply two complex numbers in general?
  2. What does i2=1i^2 = -1 represent, and why is it important in complex numbers?
  3. Can the distributive property be applied to any two complex numbers?
  4. What is the geometric interpretation of multiplying two complex numbers?
  5. How do you find the magnitude and phase of the resulting complex number?

Tip: When multiplying complex numbers, always remember that i2=1i^2 = -1, which is essential for simplifying terms involving ii.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Complex Numbers
Multiplication of Complex Numbers

Formulas

(a + bi)(c + di) = (ac - bd) + (ad + bc)i
i^2 = -1

Theorems

Distributive Property
FOIL Method

Suitable Grade Level

Grades 10-12