Problem 69 - Intermediate Algebra

Question

Find each of the products and express the answers in the standard form of a complex number. $$ (3+2 i)(5+4 i) $$

Step-by-Step Solution

Answer

The product is \(7 + 22i\).

Step 1: Apply the distributive property

To find the product of the two complex numbers, apply the distributive property or FOIL method. Multiply each term in the first complex number by each term in the second complex number: \[(3+2i)(5+4i) = 3 \times 5 + 3 \times 4i + 2i \times 5 + 2i \times 4i\]

Step 2: Multiply the real parts and imaginary parts

Complete the multiplication for each pair from Step 1: \[3 \times 5 = 15\]\[3 \times 4i = 12i\]\[2i \times 5 = 10i\]\[2i \times 4i = 8i^2\]

Step 3: Simplify the complex terms

Substitute \(i^2 = -1\) into the equation to simplify the result of the imaginary numbers multiplication:\[ 8i^2 = 8(-1) = -8\]

Step 4: Combine like terms

Combine real numbers together and imaginary numbers together: \[(15 - 8) + (12i + 10i) = 7 + 22i\]

Step 5: Write the answer in standard form

Express the result as a complex number in standard form \(a + bi\): \[7 + 22i\]

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra used to multiply a single term by multiple terms within a parenthesis. In the case of multiplying two binomials, such as complex numbers, each term in the first binomial is multiplied by every term in the second binomial.
This property ensures that you distribute the multiplication evenly across the entire expression.
For example, when we multiply the complex numbers

• distributive property allows for calculations such as:
  • 3 multiplied by 5 plus 3 multiplied by 4i
  • plus 2i multiplied by 5 and 2i multiplied by 4i.