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Chapter 2: Problem 25

Divide and express the result in standard form. $$\frac{8 i}{4-3 i}$$

Short Answer

Expert verified
The result is \(-\frac{24}{25} + \frac{32}{25}i\)

Step by step solution

01

Identify the conjugate

The conjugate of a complex number \(a + bi\) is \(a - bi\). Therefore, in this case, the conjugate of the denominator \(4 - 3i\) is \(4 + 3i\)
02

Multiply by the conjugate

Multiply both the numerator and the denominator by the conjugate of the denominator. This gives \(\frac{8i(4 + 3i)}{(4 - 3i)(4 + 3i)}\)
03

Simplify and Rearrange

Expand and simplify both the numerator and the denominator. The denominator becomes \(4^2 - (3i)^2 = 16 - -9 = 25\). The numerator becomes \(32i + 24i^2\). Remembering that \(i^2 = -1\), the numerator can be rearranged into \(32i - 24\)
04

Division and Standard Form

Divide both parts of the numerator by the denominator, giving \(\frac{-24}{25} + \frac{32}{25}i\), which is the result in standard form.

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Most popular questions from this chapter

a. Find the slant asymptote of the graph of each rational function and \(\mathbf{b} .\) Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{3}+1}{x^{2}+2 x}$$

What is a polynomial inequality?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I began the solution of the rational inequality \(\frac{x+1}{x+3} \geq 2\) by setting both \(x+1\) and \(x+3\) equal to zero.

Use transformations of \(f(x)=\frac{1}{x}\) or \(f(x)=\frac{1}{x^{2}}\) to graph each rational function. $$h(x)=\frac{1}{(x-3)^{2}}+2$$

a. Find the slant asymptote of the graph of each rational function and \(\mathbf{b} .\) Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{2}+x-6}{x-3}$$
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