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Chapter 2: Problem 86

Simplify the complex number and write it in standard form. $$(\sqrt{-2})^{6}$$

Short Answer

Expert verified
The simplified form of the complex number \((\sqrt{-2})^{6}\) in standard form is \(-8 + 0i\).

Step by step solution

01

Convert to Complex Number

Firstly, \(\sqrt{-1}\) is by definition \(i\), the basic imaginary number. Therefore, we rewrite \(\sqrt{-2}\) as \(i\sqrt{2}\) since \(\sqrt{-2}\) equals \(\sqrt{-1} * \sqrt{2}\).
02

Apply Power

Next, raise \(i\sqrt{2}\) to the 6th power. To do this, use the rule of power of a product. \((i\sqrt{2})^{6} = i^{6} * (\sqrt{2})^{6}\).
03

Evaluate Powers

Evaluate \(i^{6}\) and \((\sqrt{2})^{6}\). Remember that \(i^{1} = i\), \(i^{2} = -1\), \(i^{3} = -i\), \(i^{4} = 1\), \(i^{5} = i\), \(i^{6} = -1\), and the pattern repeats every 4. So \(i^{6} = -1\). Also, \((\sqrt{2})^{6} = (\sqrt{2})^{2*3} = (2)^{3} = 8\). So the expression simplifies to \((-1)*8 = -8\).
04

Write in Standard Form

The final step involves writing the answer in standard form \(a + ib\). Since this value generated from the power expression is a real number, it can be expressed as \(-8 + 0i\).

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Most popular questions from this chapter

Write the polynomial as the product of linear factors and list all the zeros of the function. $$f(y)=y^{4}-256$$

Use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of the function. $$h(x)=2 x^{4}-3 x+2$$

Find all real zeros of the function. $$g(x)=3 x^{3}-2 x^{2}+15 x-10$$

(a) Find the interval(s) for \(b\) such that the equation has at least one real solution and (b) write a conjecture about the interval(s) based on the values of the coefficients. $$x^{2}+b x-4=0$$

Use the information in the table to answer each question. $$\begin{array}{|c|c|} \hline \text { Interval } & \text { Value of } f(x) \\\\\hline(-\infty,-2) & \text { Positive } \\\\\hline(-2,1) & \text { Negative } \\\\\hline(1,4) & \text { Negative } \\\\\hline(4, \infty) & \text { Positive } \\\\\hline\end{array}$$ (a) What are the three real zeros of the polynomial function \(f ?\) (b) What can be said about the behavior of the graph of \(f\) at \(x=1 ?\) (c) What is the least possible degree of \(f ?\) Explain. Can the degree of \(f\) ever be odd? Explain. (d) Is the leading coefficient of \(f\) positive or negative? Explain. (e) Sketch a graph of a function that exhibits the behavior described in the table.
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