Math Problem Statement

Multiply and simplify the following complex numbers:

[(\pink{2-2i}) \cdot (\blue{4-4i})]

Solution

To multiply and simplify the complex numbers (22i)(44i)(2 - 2i) \cdot (4 - 4i), we'll use the distributive property (FOIL method).

Let's go through the steps:

  1. First: Multiply the first terms from each binomial: 24=82 \cdot 4 = 8

  2. Outer: Multiply the outer terms: 2(4i)=8i2 \cdot (-4i) = -8i

  3. Inner: Multiply the inner terms: 2i4=8i-2i \cdot 4 = -8i

  4. Last: Multiply the last terms: 2i(4i)=8i2-2i \cdot (-4i) = 8i^2

Recall that i2=1i^2 = -1, so: 8i2=8(1)=88i^2 = 8(-1) = -8

Now, add up all the terms: 88i8i88 - 8i - 8i - 8

Combine the real and imaginary parts: (88)+(8i8i)=016i(8 - 8) + (-8i - 8i) = 0 - 16i

Thus, the simplified result of multiplying (22i)(44i)(2 - 2i) \cdot (4 - 4i) is: 16i\boxed{-16i}

Would you like to see further details or have any questions?

Here are 5 related questions:

  1. What happens when you square a complex number?
  2. How do you divide complex numbers?
  3. How do you determine if two complex numbers are equal?
  4. What are the real and imaginary parts of a complex number?
  5. How do you graph a complex number on the complex plane?

Tip: When multiplying complex numbers, always simplify i2i^2 to 1-1 to get the correct real and imaginary components.

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Multiplication of Complex Numbers
Imaginary Unit

Formulas

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Theorems

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Suitable Grade Level

Grades 11-12