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Chapter 2: Problem 76

Determine whether the statement is true or false. Justify your answer. $$i^{44}+i^{150}-i^{74}-i^{109}+i^{61}=-1$$

Short Answer

Expert verified
The statement is false.

Step by step solution

01

Simplify each power of i

Since powers of \( i \) repeat every 4, we can simplify by taking each exponent modulo 4. The modulo operation gives the remainder when the number is divided by 4. So, \( i^{44} \) is equivalent to \( i^{0} \). Similarly for other terms, \( i^{150} = i^{2}, i^{74} = i^{2}, i^{109} = i^{1}, i^{61} = i^{1} \).
02

Substitute the values of i

Substitute the values of \( i^0, i^1, i^2 \) into the equation. \(i^0 = 1, i^1 = i, i^2 = -1\). So the expression becomes \(1 + (-1) - (-1) - i + i \).
03

Simplification and checking

Simplify the expression, which results in \( 1 \). This is not equal to \( -1 \), so the statement is false.

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Most popular questions from this chapter

The numbers \(D\) of doctorate degrees (in thousands) awarded to female students from 1991 through 2012 in the United States can be approximated by the model $$D=0.0743 t^{2}+0.628 t+42.61,0 \leq t \leq 22$$ where \(t\) is the year, with \(t=1\) corresponding to \(1991 .\) (Source: U.S. National Center for Education Statistics) (a) Use a graphing utility to graph the model. (b) Use the zoom and trace features to find when the number of degrees was between 50 and 60 thousand. (c) Algebraically verify your results from part (b).

Operations with Rational Expressions Simplify the expression. $$25 y^{2} \div \frac{x y}{5}$$

Find the domain of \(x\) in the expression. $$\sqrt[4]{2 x^{2}+4 x+3}$$

Solving an Equation Involving Fractions Find all solutions of the equation. Check your solutions. $$\frac{1}{t^{2}}+\frac{8}{t}+15=0$$

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. $$y=|x-2|-3$$
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