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Chapter 9: Problem 40

Write each quotient in standard form. $$ \frac{21+i}{4+i} $$

Short Answer

Expert verified
5 - i

Step by step solution

01

- Identify the problem

To write the quotient \( \frac{21+i}{4+i} \) in standard form.
02

- Multiply by the conjugate of the denominator

To eliminate the imaginary part in the denominator, multiply both the numerator and the denominator by the conjugate of the denominator, \(4-i\): \[\frac{21+i}{4+i} \times \frac{4-i}{4-i} = \frac{(21+i)(4-i)}{(4+i)(4-i)}\]
03

- Perform the multiplication in the numerator

Expand \((21+i)(4-i)\): \[(21)(4) + (21)(-i) + (i)(4) + (i)(-i) = 84 - 21i + 4i - i^2\] Since \(i^2 = -1\), substitute to get: \[(84 - 21i + 4i + 1) = 85 - 17i\]
04

- Perform the multiplication in the denominator

Expand \((4+i)(4-i)\): \[(4)(4) + (4)(-i) + (i)(4) + (i)(-i) = 16 - 4i + 4i - i^2\]. Since \(i^2 = -1\), substitute to get: \(16 + 1 = 17\)
05

- Write the result in standard form

The fraction now is \[\frac{85-17i}{17}\]. Divide each term of the numerator by the denominator: \[\frac{85}{17} - \frac{17i}{17} = 5 - i\]. Therefore, the standard form is \(5 - i\).

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Key Concepts

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

Conjugate
In complex numbers, the conjugate is a crucial concept. It helps to simplify complex division and other operations. Each complex number has a conjugate. To find the conjugate, you change the sign of the imaginary part.

For example, if you have a complex number like \(4+i\), its conjugate is \(4-i\).

Why is this useful? When you multiply a complex number by its conjugate, you eliminate the imaginary part. This tactic converts a complex number into a real number, making calculations simpler.

In our exercise, to simplify \( \frac{21+i}{4+i} \), we multiply by the conjugate of the denominator: \(4-i\). This helps us eliminate the imaginary unit in the denominator.
Standard Form
Standard form for complex numbers is written as \(a+bi\), where \(a\) is the real part, and \(b\) is the imaginary part.

In the exercise, our aim is to express the quotient in this format.

Starting from \( \frac{21+i}{4+i} \), we multiply by the conjugate to get:
\( \frac{(21+i)(4-i)}{(4+i)(4-i)} \).

After performing the multiplications and simplifying, we find:
\( \frac{85-17i}{17} = 5 - i \).

Thus, \(5 - i\) is in standard form, with \(5\) as the real part and \(-1\) as the imaginary part.
Imaginary Unit
The imaginary unit, represented by \(i\), is the foundation of complex numbers. It is defined as \(i = \sqrt{-1}\). One key property is that \(i^2 = -1\).

This property plays a fundamental role in calculations involving complex numbers.

In our example, when we expanded \( (21+i)(4-i) \) and \( (4+i)(4-i) \), we encountered terms like \( -i^2 \).

Since \(i^2 = -1\), substituting helped us convert these terms into real numbers. This conversion is crucial for simplifying complex expressions and achieving the standard form.

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Most popular questions from this chapter

Use the quadratic formula to solve each equation. (a) Give solutions in exact form, and (b) use a calculator to give solutions correct to the nearest thousandth. $$ x^{2}=1+x $$

Solve each problem. A rule for estimating the number of board feet of lumber that can be cut from a log depends on the diameter of the log. To find the diameter \(d\) required to get 9 board feet of Iumber, we use the equation $$ \left(\frac{d-4}{4}\right)^{2}=9 $$ Solve this equation for \(d .\) Are both answers reasonable?

Use a calculator with a square root key to solve each equation. Round your answers to the nearest hundredth. \((r-3.91)^{2}=9.28\)

Evaluate each expression for \(x=3\). $$ 2 x^{2}-x+1 $$

Simplify all radicals, and combine like terms. Express fractions in lowest terms. \(\frac{12-\sqrt{27}}{9}\)
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