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Chapter 2: Problem 85

Simplify the complex number and write it in standard form. $$(\sqrt{-72})^{3}$$

Short Answer

Expert verified
-1024\sqrt{2}i

Step by step solution

01

Simplify the square root

The square root of a negative number can be obtained by taking the square root of the absolute value of the number and appending \(i\). Thus, \(\sqrt{-72} = \sqrt{72} \cdot i = 8\sqrt{2} \cdot i\). \(\sqrt{72}\) is simplified to \(8\sqrt{2}\) by breaking down \(72\) into \(4 * 18\), then \(2 * 2 * 9 * 2\), and taking the square root of \(64\) to end up with \(8\sqrt{2}\).
02

Cube the complex number

Now that \(\sqrt{-72}\) is simplified to a form \(8\sqrt{2} \cdot i\), the cube of this expression should be calculated. Thus, \((8\sqrt{2} \cdot i)^{3}= (8\sqrt{2})^3 \cdot i^3 = 512\sqrt{8} \cdot i^3\). The square root of \(8\) can be further simplified to \(2\sqrt{2}\), resulting in \(1024\sqrt{2} \cdot i^3\).
03

Simplify further by taking into account the properties of \(i\)

Since \(i^2 = -1\), \(i^3 = i^2 \cdot i = -1 \cdot i = -i\). Substituting \(-i\) for \(i^3\) in the previous expression, \(1024\sqrt{2} \cdot -i = -1024\sqrt{2} i\), the expression is now in standard form for complex numbers, which is \(a + bi\), where \(a\) and \(b\) are real numbers.

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Most popular questions from this chapter

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.

Cube each complex number. (a) \(-1+\sqrt{3} i\) (b) \(-1-\sqrt{3} i\)

Write the polynomial as the product of linear factors and list all the zeros of the function. $$f(x)=x^{4}-16$$

Decide whether the statement is true or false. Justify your answer. It is possible for a third-degree polynomial function with integer coefficients to have no real zeros.

Write the polynomial as the product of near factors and list all the zeros of the function. $$f(x)=x^{4}+29 x^{2}+100$$
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