Math Problem Statement

4-i / 3+2i

Solution

The expression you've provided is:

4i3+2i\frac{4 - i}{3 + 2i}

To simplify this expression, we can multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of 3+2i3 + 2i is 32i3 - 2i.

So, let's do that:

(4i)(32i)(3+2i)(32i)\frac{(4 - i) \cdot (3 - 2i)}{(3 + 2i) \cdot (3 - 2i)}

Step 1: Multiply the numerator

(4i)(32i)=4342ii3+i2i(4 - i)(3 - 2i) = 4 \cdot 3 - 4 \cdot 2i - i \cdot 3 + i \cdot 2i =128i3i2i2= 12 - 8i - 3i - 2i^2

Since i2=1i^2 = -1, the expression simplifies to: =128i3i+2=1411i= 12 - 8i - 3i + 2 = 14 - 11i

Step 2: Multiply the denominator

(3+2i)(32i)=32(2i)2=94i2(3 + 2i)(3 - 2i) = 3^2 - (2i)^2 = 9 - 4i^2 =94(1)=9+4=13= 9 - 4(-1) = 9 + 4 = 13

Final Answer:

4i3+2i=1411i13\frac{4 - i}{3 + 2i} = \frac{14 - 11i}{13}

This can be separated into real and imaginary parts: 14131113i\frac{14}{13} - \frac{11}{13}i

So, the simplified form of 4i3+2i\frac{4 - i}{3 + 2i} is: 14131113i\frac{14}{13} - \frac{11}{13}i

Would you like to see more details or have any other questions?


Here are 8 related questions:

  1. How do you find the conjugate of a complex number?
  2. What is the significance of multiplying by the conjugate in complex number division?
  3. Can you simplify 2+3i14i\frac{2 + 3i}{1 - 4i} using the same method?
  4. What happens if you multiply two complex conjugates together?
  5. How do you add and subtract complex numbers?
  6. How do you multiply two complex numbers without using their conjugates?
  7. What is the modulus of a complex number?
  8. How do you find the argument of a complex number?

Tip: When working with complex numbers, always remember that i2=1i^2 = -1. This fundamental property simplifies many expressions involving imaginary numbers.

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Conjugates
Division of Complex Numbers

Formulas

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Theorems

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Suitable Grade Level

Advanced High School