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Chapter 11: Problem 50

Multiply and simplify. $$\left(\frac{1}{3}-\frac{4}{3} i\right)\left(\frac{3}{4}+\frac{2}{3} i\right)$$

Short Answer

Expert verified
\(\left(\frac{1}{3}-\frac{4}{3}i\right)\left(\frac{3}{4}+\frac{2}{3}i\right) = \frac{41}{36} - \frac{7}{9}i\)

Step by step solution

01

Identify the complex numbers

We are given two complex numbers in the form of a fraction: \(A = \frac{1}{3}-\frac{4}{3}i\) and \(B = \frac{3}{4}+\frac{2}{3}i\)
02

Multiply the complex numbers

We will use the formula for multiplying complex numbers: \((a + bi)(c + di) = (ac - bd) + (ad + bc)i\) Using this formula, we will multiply A and B: \((\frac{1}{3}-\frac{4}{3}i)(\frac{3}{4}+\frac{2}{3}i) = \left(\frac{1}{3}\cdot\frac{3}{4}-(-\frac{4}{3}\cdot\frac{2}{3})\right) + \left(\frac{1}{3}\cdot\frac{2}{3} + (-\frac{4}{3}\cdot\frac{3}{4})\right)i\)
03

Calculate and simplify the result

Now, we will simplify the real and imaginary parts of the result: Real part: \(\frac{1}{3}\cdot\frac{3}{4} + \frac{4}{3}\cdot\frac{2}{3} = \frac{1}{4} + \frac{8}{9}\) Imaginary part: \(\frac{1}{3}\cdot\frac{2}{3} - \frac{4}{3}\cdot\frac{3}{4} = \frac{2}{9} - 1\) Now, let's find the common denominator and add the terms in the real part: \(\frac{1}{4} + \frac{8}{9} = \frac{9}{36} + \frac{32}{36} = \frac{41}{36}\) In the imaginary part, the common denominator is already present, so we can directly subtract the terms: \(\frac{2}{9} - 1 = \frac{2}{9} - \frac{9}{9} = -\frac{7}{9}\)
04

Write the final answer

Combine the simplified real and imaginary parts to get the final answer: \(\left(\frac{1}{3}-\frac{4}{3}i\right)\left(\frac{3}{4}+\frac{2}{3}i\right) = \frac{41}{36} - \frac{7}{9}i\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Imaginary Unit
The imaginary unit, denoted as \(i\), is a fundamental concept when dealing with complex numbers. It is defined as the square root of -1, implying that \(i^2 = -1\). This special property provides a way to extend the real number system to include solutions to equations that don't have real solutions, such as \(x^2 + 1 = 0\).
When multiplying complex numbers, it's essential to handle the powers of \(i\) correctly.
  • \(i^2\) results in -1, affecting how terms in complex number expressions simplify.
  • When multiplying expressions that include \(i\), keep track of these powers to simplify properly.
Working with \(i\) might seem unusual at first, but it opens up a wide range of possibilities, especially in fields such as engineering and physics.
It's a stepping stone for handling and operating on complex numbers smoothly.
Complex Number Simplification
Simplification of complex numbers includes organizing both the real and imaginary parts into a clearer, most reduced form. A complex number is typically expressed as \(a + bi\), where \(a\) represents the real part and \(b\) is the coefficient of the imaginary part.
In multiplication, like our given expression \((\frac{1}{3} - \frac{4}{3}i)(\frac{3}{4} + \frac{2}{3}i)\), we must apply the distributive property, often referred to as 'FOIL' for two binomials:
  • First: Multiply the first terms in the binomial expressions.
  • Outside: Multiply the outer terms.
  • Inside: Multiply the inner terms.
  • Last: Multiply the last terms.
After multiplying, we simplify:
  • Combine like terms (real and imaginary separately).
  • Apply \(i^2 = -1\) where necessary to rewrite imaginary parts correctly.
Once simplified, you'll express it back in standard form \(a + bi\), ensuring that both parts are as simple as possible.
Multiplying Fractions
When multiplying fractions, the process is straightforward and is an essential skill for simplifying expressions with complex numbers. The multiplication involves:
  • Multiplying the numerators to get the new numerator.
  • Multiplying the denominators to get the new denominator.
In this complex number scenario, the fractions appear frequently in both the real and imaginary components. For instance, when multiplying \(\frac{1}{3}\) and \(\frac{3}{4}\), you directly obtain \(\frac{1 \cdot 3}{3 \cdot 4} = \frac{1}{4}\).
Be mindful of common denominators when adding or subtracting the resulting fractions. To combine fractions properly:
  • Find a common denominator if you need to add or subtract fractions.
  • Convert the fractions accordingly.
  • Perform the addition or subtraction.
With practice, handling fractions and incorporating them into simplifying complex numbers becomes second nature.

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