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Chapter 2: Problem 65

Perform the operation and write the result in standard form. $$\frac{i}{3-2 i}+\frac{2 i}{3+8 i}$$

Short Answer

Expert verified
The result in standard form of the given operation \(\frac{i}{3-2 i}+\frac{2 i}{3+8i}\) is \(0.15 - 0.14i\)

Step by step solution

01

Rationalize the denominators

In order to rationalize the denominator, multiply both the numerator and denominator by the conjugate of the denominator of the fractions. \nThe conjugate of \(3 - 2i\) is \(3 + 2i\) and of \(3 + 8i\) is \(3 - 8i\). Hence, \(\frac{i}{3-2 i} \times \frac{3+2i}{3+2i} + \frac{2 i}{3+8 i} \times \frac{3-8i}{3-8i}\)
02

Simplify the expressions

Now, simplify the expressions obtained in step 1 using the rule \(i^2=-1\) and the property \((a + bi)(a - bi)=a^2 + b^2\).\nAfter simplification, you will get \(\frac{3i+2}{13} + \frac{6-16i}{73}\)
03

Combine the fractions

Next, write these complex numbers in standard form: \(\frac{2}{13} + \frac{3i}{13} +\frac{6}{73} -\frac{16i}{73}\)
04

Combine Like Terms

After simplifying, combine like terms.\nThe answer will be: \(0.15 - 0.14i\) which is the standard form of the result.

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Most popular questions from this chapter

Use a graphing utility to graph the equation and graphically approximate the values of \(x\) that satisfy the specified inequalities. Then solve each inequality algebraically. Equation \(y=\frac{3 x}{x-2}\) Inequalities (a) \(y \leq 0\) (b) \(y \geq 6\)

Operations with Rational Expressions Simplify the expression. $$\frac{2}{z+2}-\left(3-\frac{2}{z}\right)$$

use the models below, which approximate the numbers of Bed Bath \(\&\) Beyond stores \(B\) and Williams-Sonoma stores \(W\) for the years 2000 through \(2013,\) where \(t\) is the year, with \(t=0\) corresponding to 2000. (Sources: Bed Bath \(\&\) Beyond, Inc.; Williams-Sonoma, Inc.) Bed Bath \& Beyond: $$B=86.5 t+342,0 \leq t \leq 13$$ Williams-Sonoma: $$W=-2.92 t^{2}+52.0 t+381,0 \leq t \leq 13$$. Solve the inequality \(W(t) \leq 600 .\) Explain what the solution of the inequality represents.

Twenty years ago, your parents deposited \(\$ 3000\) in a long-term investment in which interest was compounded biannually. Today, the value of the investment is \(\$ 8055.19 .\) What is the annual interest rate for this investment?

use the models below, which approximate the numbers of Bed Bath \(\&\) Beyond stores \(B\) and Williams-Sonoma stores \(W\) for the years 2000 through \(2013,\) where \(t\) is the year, with \(t=0\) corresponding to 2000. (Sources: Bed Bath \(\&\) Beyond, Inc.; Williams-Sonoma, Inc.) Bed Bath \& Beyond: $$B=86.5 t+342,0 \leq t \leq 13$$ Williams-Sonoma: $$W=-2.92 t^{2}+52.0 t+381,0 \leq t \leq 13$$. Solve the inequality \(B(t) \geq 900 .\) Explain what the solution of the inequality represents.
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