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Chapter 2: Problem 22

Divide and express the result in standard form. $$\frac{3}{4+i}$$

Short Answer

Expert verified
The result of dividing, expressed in standard for is \(\frac{12}{17}-\frac{3}{17}i\).

Step by step solution

01

Write down the Complex number

The given fraction is \(\frac{3}{4+i}\).
02

Rationalize the denominator

As the denominator has imaginary number \(i\), it should be rationalized by multiplying the numerator and the denominator by the conjugate of the denominator, which is \(4-i\). \nSo, we get:\n\(\frac{3}{4+i} * \frac{4-i}{4-i} = \frac{3(4-i)}{(4+i)(4-i)}\)
03

Perform multiplication in numerator

Multiply the terms in the numerator to get: \n\n\(= \frac{12-3i}{(4+i)(4-i)}\)
04

Perform multiplication in denominator

Multiply the terms in the denominator. Remember that \(i^2=-1\) \n\n\(=\frac{12-3i}{4^2 - i^2}\)\(=\frac{12-3i}{16+1}\)\(=\frac{12-3i}{17}\)
05

Simplify the expression

Now simplify the expression by dividing each term in the numerator by the denominator: \n\n\(=\frac{12}{17}-\frac{3}{17}i\)

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Most popular questions from this chapter

The equation for \(f\) is given by the simplified expression that results after performing the indicated operation. Write the equation for \(f\) and then graph the function. $$\frac{2}{x^{2}+3 x+2}-\frac{4}{x^{2}+4 x+3}$$

Use transformations of \(f(x)=\frac{1}{x}\) or \(f(x)=\frac{1}{x^{2}}\) to graph each rational function. $$g(x)=\frac{1}{x+1}-2$$

You drive from your home to a vacation resort 600 miles away. You return on the same highway. The average velocity on the return trip is 10 miles per hour slower than the average velocity on the outgoing trip. Express the total time required to complete the round trip, \(T\), as a function of the average velocity on the outgoing trip, \(x .\)

The equation for \(f\) is given by the simplified expression that results after performing the indicated operation. Write the equation for \(f\) and then graph the function. $$\frac{1-\frac{3}{x+2}}{1+\frac{1}{x-2}}$$

Solve each inequality using a graphing utility. $$2 x^{2}+5 x-3 \leq 0$$
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