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Chapter 2: Problem 71

Simplify the complex number and write it in standard form. $$\frac{1}{i^{3}}$$

Short Answer

Expert verified
-i

Step by step solution

01

Simplify the Expression

Start by simplifying the expression \(i^{3}\). Remember that the imaginary unit \(i\) has the property that \(i^{2} = -1\). That means \(i^{3} = i^{2} \cdot i = -i\). Therefore, the given expression becomes \(\frac{1}{-i}\).
02

Rationalize the Denominator

Next, rationalize the denominator by multiplying both the numerator and the denominator by the complex conjugate of -i which is i: \(\frac{1}{-i} \cdot \frac{i}{i} = -i\).
03

Write in Standard Form

The complex number -i can also be written as \(0 - 1i\) which is a standard form of a complex number \(a + bi\), where \(a = 0\) and \(b = -1\).

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Most popular questions from this chapter

Completely factor the expression over the real numbers. $$x^{3}-5 x^{2}-14 x$$

Find the domain of \(x\) in the expression. $$\sqrt{6 x+15}$$

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any \(x\) -intercepts of the graph, (c) set \(y=0\) and solve the resulting equation, and (d) compare the result of part (c) with the \(x\) -intercepts of the graph. $$y=\sqrt{7 x+36}-\sqrt{5 x+16}-2$$

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any \(x\) -intercepts of the graph, (c) set \(y=0\) and solve the resulting equation, and (d) compare the result of part (c) with the \(x\) -intercepts of the graph. $$y=2 x-\sqrt{15-4 x}$$

Solving an Equation Involving Fractions Find all solutions of the equation. Check your solutions. $$6\left(\frac{s}{s+1}\right)^{2}+5\left(\frac{s}{s+1}\right)-6=0$$
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