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Chapter 1: Problem 62
Write each quotient in the form \(a+b d\) $$\frac{4+2 i}{2-3 i}$$
Short Answer
Expert verified
\(\frac{2}{13} + \frac{16}{13}i\)
Step by step solution
01
Identify the expression
The given expression is \(\frac{4+2i}{2-3i}\). We need to write this quotient in the form \(a + b i\).
02
Multiply the numerator and denominator by the conjugate
Multiply both the numerator and the denominator by the conjugate of the denominator, which is \(2+3i\). The expression becomes: \(\frac{(4+2i)(2+3i)}{(2-3i)(2+3i)}\).
03
Expand the numerator
Expand \((4+2i)(2+3i)\) using the distributive property (FOIL method): \(4 \times 2 + 4 \times 3i + 2i \times 2 + 2i \times 3i\). This simplifies to \(8 + 12i + 4i + 6i^2\).
04
Simplify the numerator
Combine like terms in the numerator: \(8 + 16i + 6i^2\). Since \(i^2 = -1\), replace \(6i^2\) with \(6 \times -1 = -6\). Therefore, the simplified numerator is \(8 + 16i - 6 = 2 + 16i\).
05
Calculate the denominator
The denominator is \((2-3i)(2+3i)\). Using the difference of squares formula: \(a^2 - b^2\) where \(a = 2\) and \(b = 3i\), we get \(2^2 - (3i)^2\), which simplifies to \(4 - 9i^2\). Since \(i^2 = -1\), \(4 - 9(-1) = 4 + 9 = 13\).
06
Write the simplified expression
The expression becomes \(\frac{2 + 16i}{13}\). This can be rewritten as two separate fractions: \(\frac{2}{13} + \frac{16}{13}i\).
07
Final answer
Thus, the expression \(\frac{4 + 2i}{2 - 3i}\) in the form \(a + bi\) is \(\frac{2}{13} + \frac{16}{13}i\).
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
complex conjugate
Complex conjugates are crucial when dealing with the division of complex numbers.
They allow us to eliminate the imaginary unit from the denominator.
To form a complex conjugate, you change the sign of the imaginary part of a complex number.
Consider a complex number 2 + 3i. Its complex conjugate is 2 - 3i.
This property is particularly useful for simplifying expressions.
When dividing by a complex number, you multiply both the numerator and the denominator by the conjugate of the denominator.
In our example: to divide 4 + 2i by 2 - 3i, we multiply both by 2 + 3i.
This changes the denominator to a real number, simplifying our division process.
distributive property
The distributive property helps us to multiply complex numbers efficiently.
It states that a(b + c) = ab + ac .
When both the numerator and the denominator were multiplied by the complex conjugate, we expanded the product using the distributive property.
This involved calculating terms like:
- 4 * 2
- 4 * 3i
- 2i * 2
- 2i * 3i
.
Using the distributive property ensures all terms are accounted for.
The combination of these smaller products into a single expression makes the process easier to manage and follow.
difference of squares
The difference of squares is a special algebraic formula,
written as a^2 - b^2.
It occurs frequently in operations involving complex numbers, especially in our scenario.
When multiplying (2 - 3i)(2 + 3i), we convert it into this form: 2^2 - (3i)^2.
Notice that (a^2 - b^2) = (a - b)(a + b).
This helps to eliminate the imaginary parts, transforming our denominator into a real number.
Specifically, 4 - (-9) simplifies to 4 + 9, yielding 13 as the final denominator value.
This simplifies the overall expression significantly and allows us to separate the real and imaginary parts.
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