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Chapter 9: Problem 56

Rationalize the denominator. Write all answers in a + bi form. $$\frac{4+5 i}{4-5 i}$$

Short Answer

Expert verified
-9/41 + (40/41)i

Step by step solution

01

Identification of the conjugate

The conjugate of any complex number a + bi is a - bi. So, the conjugate of the denominator 4 - 5i is 4 + 5i.
02

Multiply by the conjugate

Multiply the numerator and the denominator of the fraction by 4 + 5i, the conjugate of the denominator. Doing this gives the new term \(\frac{(4+5i)(4+5i)}{(4-5i)(4+5i)}\).
03

Expand and Simplify

Expand the expressions in the formula. For the numerator, use the formula (a+b)^2 = a^2 + 2ab + b^2 to obtain (4^2 + 2*4*5i + (5i)^2) = 16 + 40i - 25. For the denominator, use the formula (a - b)(a + b) = a^2 - b^2 to obtain (4^2 - (5i)^2) = 16 + 25 = 41. So, now, we have \(\frac{16 + 40i - 25}{41}\).
04

Write in the form a+bi

Simplify the numerator and rewrite in the form a+bi. The simplified numerator is -9 + 40i. Therefore, the fraction becomes \(\frac{-9 + 40i}{41}\).
05

Final Solution

To write this in a+bi form, split the fraction: -9/41 + (40/41)i. This is the final answer.

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Key Concepts

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

Rationalizing the Denominator
Rationalizing the denominator is a fundamental strategy used in simplifying expressions with complex numbers. When we have a fraction with a complex number in the denominator, it's preferable to "rationalize" it. This means transforming the fraction so that the denominator becomes a real number.

Here's how to rationalize the denominator when it contains a complex number:
  • Identify the complex conjugate of the denominator. The complex conjugate reverses the sign of the imaginary part of the number.
  • Multiply both the numerator and the denominator of the fraction by this conjugate.
  • Through this multiplication, the imaginary parts cancel out in the denominator.
Applying this process provides a fraction with a real number in the denominator, making further operations much simpler.
Complex Conjugate
The complex conjugate is a vital concept when dealing with complex numbers. A complex number is typically in the form of \(a + bi\), and its complex conjugate is \(a - bi\).

Why is the complex conjugate important, especially when rationalizing denominators? Here鈥檚 why:
  • Using the conjugate helps to eliminate the imaginary part of the denominator, turning it into a real number.
  • When a complex number is multiplied by its conjugate, the result is always a real number, specifically \(a^2 + b^2\).
  • This process leverages the difference of squares \((a - b)(a + b) = a^2 - b^2\).
By multiplying using the complex conjugate, we not only simplify operations but ensure the expression is in a more workable form.
a + bi Form
Writing complex numbers in the standard form \(a + bi\) means expressing them in terms of a real part (\(a\)) and an imaginary part (\(bi\)).

Here's why the \(a + bi\) form is used:
  • It provides a consistent way to easily identify both the real and imaginary parts of a complex number.
  • It simplifies the process of performing arithmetic operations like addition or subtraction on complex numbers.
  • Many mathematical concepts, including polar and exponential forms, build upon this basic representation.
In the exercise, after rationalizing the denominator and simplifying the fraction, the complex number is split into its real and imaginary components to fit \(a + bi\) form: \(-\frac{9}{41} + \frac{40}{41}i\). This clear separation helps in understanding the nature of complex numbers in any calculations that follow.

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Most popular questions from this chapter

Use the quadratic formula to determine whether there are real number solutions to each equation. SEE EXAMPLE 4. (OBJECTIVE 2) $$x^{2}+5=2 x$$

Set up an equation to solve. An electronics firm has found that its revenue for manufacturing and selling \(x\) television sets is given by the formula \(R=-\frac{1}{6} x^{2}+450 x\). How much revenue will be earned by manufacturing 600 television sets?

Simplify: \(i^{-1}\)

Simplify. Write all answers in a \(+\) bi form. $$\frac{3+\sqrt{-2}}{2+\sqrt{-5}}$$

Solve each equation. Write the answer in bi or a \(+\) bi form. $$3 x^{2}-4 x+2=0$$
See all solutions

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