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Chapter 8: Problem 62

For Exercises 49-64, write each quotient in standard form. $$ \frac{7+4 i}{9-3 i} $$

Short Answer

Expert verified
The quotient in standard form is \( \frac{17}{30} + \frac{19}{30}i \).

Step by step solution

01

Write the Original Expression

The given expression is \( \frac{7+4i}{9-3i} \). Our goal is to write this quotient in standard form, which is \( a + bi \), where \( a \) and \( b \) are real numbers.
02

Multiply by the Conjugate

To eliminate the imaginary unit \( i \) from the denominator, multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of \( 9-3i \) is \( 9+3i \).Perform the multiplication:\[ \frac{(7+4i)(9+3i)}{(9-3i)(9+3i)} \]
03

Simplify the Denominator

Simplify the denominator using the difference of squares formula, \( a^2 - b^2 \):\[ (9)^2 - (3i)^2 = 81 - 9(-1) = 81 + 9 = 90 \]
04

Expand the Numerator

Expand the expression \((7+4i)(9+3i)\) using distributive property:\[ 7 \cdot 9 + 7 \cdot 3i + 4i \cdot 9 + 4i \cdot 3i \= 63 + 21i + 36i + 12i^2 \= 63 + 57i + 12(-1) = 63 + 57i - 12 = 51 + 57i \]
05

Divide Real and Imaginary Components

Now, separate the real part and the imaginary part of the result:The expression becomes \( \frac{51 + 57i}{90} \).This can be written as: \\( \frac{51}{90} + \frac{57}{90}i \).
06

Simplify Each Component

Simplify each part: \( \frac{51}{90} = \frac{17}{30} \) and \( \frac{57}{90} = \frac{19}{30} \).Thus, the expression in standard form is \( \frac{17}{30} + \frac{19}{30}i \).

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Key Concepts

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

Standard Form
When dealing with complex numbers, the standard form is a way of expressing them with both real and imaginary parts clearly separated. A complex number in standard form looks like this:
  • The formula: \( a + bi \)
  • Where \( a \) represents the real part of the number.
  • And \( b \) is the coefficient of the imaginary part (with the imaginary unit denoted as \( i \)).
Being able to present a complex number in standard form helps with calculations and understanding.Converting complex numbers into standard form often involves algebraic manipulations, such as multiplying by the conjugate when dealing with quotients.
Conjugate
The conjugate of a complex number is essential in simplifying expressions that have imaginary components. If you have a complex number such as \( a + bi \), its conjugate is \( a - bi \). By changing the sign of the imaginary part, we use the conjugate to eliminate the imaginary unit from the denominator of a fraction. When you multiply a number by its conjugate, it results in a real number. This outcome is because:
  • The product \( (a + bi)(a - bi) \) uses the formula \( a^2 - b^2i^2 \).
  • Since \( i^2 = -1 \), the expression simplifies to \( a^2 + b^2 \).
This technique enables the transformation of fractions with complex numbers in the denominator into standard form.
Imaginary Unit
Imaginary numbers revolve around the imaginary unit, denoted as \( i \). The defining property of the imaginary unit is that \( i^2 = -1 \). This property allows you to work with numbers that involve square roots of negative values. In practical terms:
  • Any complex number can be expressed in terms of \( i \), as in the form \( a + bi \).
  • Operations including multiplication, division, and addition with complex numbers are handled using this unit.
  • While real numbers appear along the real number line, complex numbers occupy a plane needing both real and imaginary axes.
Understanding the imaginary unit is crucial for fully grasping how complex numbers function and for performing operations with them.
Real and Imaginary Components
A complex number is made up of two parts: the real part and the imaginary part.
  • The real part is denoted by \( a \) and is a component that does not involve \( i \).
  • The imaginary component is denoted by \( b \), involved with the unit \( i \).
These components allow complex numbers to be expressed in the form \( a + bi \).When simplifying or rewriting complex numbers during operations, identifying these components is crucial. For example, in quotients:
  • You start by separating real and imaginary parts after eliminating \( i \) from the denominator.
  • The result can be simplified, giving each part its own distinct expression.
  • This leads to a clear standard form, such as \( \frac{17}{30} + \frac{19}{30}i \) seen in the exercise solution.
Recognizing these components helps in interpreting the complex number, allowing it to be utilized in further mathematical contexts.

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