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Chapter 2: Problem 83

Perform the indicated operations and write the result in standard form. $$\frac{8}{1+\frac{2}{i}}$$

Short Answer

Expert verified
The result in standard form is: \( \frac{8}{5} - \frac{16}{5}i \)

Step by step solution

01

Simplify the Inner Complex Fraction

1 + \frac{2}{i} can be simplified by multiplying both the numerator and denominator by 'i' to get \(1+ 2i\)
02

Simplify the Overall Fraction

Next, replace the denominator of the main fraction with the result from step 1 to get \(\frac{8}{1+2i}\). To get this fraction into standard complex form, multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \(1 + 2i\) is \(1 - 2i\), so the fraction becomes \(\frac{8(1-2i)}{(1+2i)(1-2i)}\).
03

Calculate the Numerator and Denominator Separately

First calculate the numerator: \( 8(1-2i) = 8 - 16i \). Then, calculate the denominator \( (1+2i)(1-2i) = 1 - 4i^2 \), remembering that \( i^2 = -1. \)
04

Complete the Simplification

Simplify: \( \frac{8 - 16i}{1 - 4(-1)} = \frac{8 - 16i}{1 + 4} = \frac{8 - 16i}{5} = \frac{8}{5} - \frac{16}{5}i \). This is the final result in standard form.

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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. $$\frac{x-\frac{1}{x}}{x+\frac{1}{x}}$$

If \(S=\frac{k A}{P},\) find the value of \(k\) using \(A=60,000, P=40,\) and \(S=12,000\).

Solve each inequality using a graphing utility. $$\frac{x-4}{x-1} \leq 0$$

Write the equation of a rational function \(f(x)=\frac{p(x)}{q(x)}\) having the indicated properties, in which the degrees of p and q are as small as possible. More than one correct function may be possible. Graph your function using a graphing utility to verify that it has the required properties. \(f\) has a vertical asymptote given by \(x=1,\) a slant asymptote whose equation is \(y=x, y\) -intercept at \(2,\) and \(x\)-intercepts at -1 and 2.

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