Math Problem Statement

Perform the indicated operation & simplify. Express the answer in terms of i (as a complex number). (9+7i)(5+2i)

Solution

We need to perform the multiplication (9+7i)(5+2i)(9 + 7i)(5 + 2i) and simplify the result.

Step 1: Apply the distributive property (FOIL method)

We distribute each term in the first binomial to each term in the second binomial:

(9+7i)(5+2i)=9(5)+9(2i)+7i(5)+7i(2i)(9 + 7i)(5 + 2i) = 9(5) + 9(2i) + 7i(5) + 7i(2i)

This simplifies to:

=45+18i+35i+14i2= 45 + 18i + 35i + 14i^2

Step 2: Simplify the result

  • Combine the real parts: 4545
  • Combine the imaginary parts: 18i+35i=53i18i + 35i = 53i
  • Simplify 14i214i^2. Since i2=1i^2 = -1, we have: 14i2=14(1)=1414i^2 = 14(-1) = -14

Now, put it all together:

45+53i14=(4514)+53i45 + 53i - 14 = (45 - 14) + 53i

This simplifies to:

31+53i31 + 53i

Final Answer:

31+53i31 + 53i

Would you like more details on this process, or do you have any other questions?

Further Questions:

  1. What is the general rule for multiplying two complex numbers?
  2. How do we handle complex conjugates in multiplication?
  3. Can we factorize the expression further?
  4. How do we add or subtract complex numbers?
  5. What is the geometric interpretation of multiplying complex numbers?

Tip: When multiplying complex numbers, always remember to use the fact that i2=1i^2 = -1, which can help

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Algebra
Multiplication of Binomials

Formulas

(a + bi)(c + di) = ac + adi + bci + bdi^2
i^2 = -1

Theorems

Properties of Complex Numbers
Distributive Property

Suitable Grade Level

Grades 9-12