Math Problem Statement

(4 + 9i)(4 - 9i)

Solution

The given problem is:

(4+9i)(49i)(4 + 9i)(4 - 9i)

To solve this, we will use the formula for multiplying complex numbers: (a+bi)(abi)=a2b2i2(a + bi)(a - bi) = a^2 - b^2i^2. Here, a=4a = 4 and b=9b = 9.

Step-by-Step Solution:

  1. Expand the expression using the difference of squares: (4+9i)(49i)=42(9i)2(4 + 9i)(4 - 9i) = 4^2 - (9i)^2
  2. Calculate 424^2: 42=164^2 = 16
  3. Calculate (9i)2(9i)^2: (9i)2=81i2(9i)^2 = 81i^2 Since i2=1i^2 = -1, this becomes: 81i2=81(1)=8181i^2 = 81(-1) = -81
  4. Combine the results: 16(81)=16+81=9716 - (-81) = 16 + 81 = 97

Final Answer:

(4+9i)(49i)=97(4 + 9i)(4 - 9i) = 97

The answer is 97, which is a real number. It should be written as 97+0i97 + 0i if you need to express it in the form a+bia + bi.

Would you like more details or have any questions?

Here are 5 related questions for further practice:

  1. Simplify (3+5i)(35i)(3 + 5i)(3 - 5i).
  2. Multiply (2+7i)(27i)(2 + 7i)(2 - 7i) and find the result.
  3. What is the product of (5+4i)(54i)(5 + 4i)(5 - 4i)?
  4. How do you multiply complex numbers in the form (a+bi)(c+di)(a + bi)(c + di)?
  5. What is the result when multiplying (62i)(6+2i)(6 - 2i)(6 + 2i)?

Tip: When multiplying complex conjugates, the imaginary parts always cancel out, resulting in a real number.

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Algebra
Difference of Squares

Formulas

(a + bi)(a - bi) = a^2 - b^2i^2

Theorems

Difference of Squares

Suitable Grade Level

Grades 10-12