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Chapter 2: Problem 89

Simplify the complex number and write it in standard form. $$(3 i)^{4}$$

Short Answer

Expert verified
The simplified complex number in standard form is 81.

Step by step solution

01

Define complex number

The given expression is \((3i)^4\). Recall that \(i\) is an imaginary unit defined by the square root of -1 (\(i = \sqrt{-1}\)) and that \(3i\) is a complex number.
02

Complex exponentiation

Raise the complex number to the specified power: \((3i)^4 = 81i^4\). The power distribitutes over the multiplication, and \(3^4 = 81\) and \(i^4 = 1\) because \(i^2 = -1\) and \((-1)^2 = 1\). Therefore, \(i^4 = (i^2)^2 = (-1)^2 = 1\).
03

Final simplification

With \(i^4\) replaced by 1, the final simplification of the expression is \(81(1) = 81\).

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Most popular questions from this chapter

Use synthetic division to verify the upper and lower bounds of the real zeros of \(f\) \(f(x)=x^{3}+3 x^{2}-2 x+1\) (a) Upper: \(x=1\) (b) Lower: \(x=-4\)

Solve the inequality. (Round your answers to two decimal places.) $$\frac{2}{3.1 x-3.7}>5.8$$

Complete the following. \begin{aligned} &i^{1}=i \quad i^{2}=-1 \quad i^{3}=-i \quad i^{4}=1\\\ &i^{5}=\quad i^{6}=\quad i^{7}=\quad i^{8}=\\\ &i^{9}=\\\ &i^{10}=\quad i^{11}=\quad i^{12}= \end{aligned} What pattern do you see? Write a brief description of how you would find \(i\) raised to any positive integer power.

Determine whether the statement is true or false. Justify your answer. The solution set of the inequality \(\frac{3}{2} x^{2}+3 x+6 \geq 0\) is the entire set of real numbers.

Find all real zeros of the function. $$f(z)=12 z^{3}-4 z^{2}-27 z+9$$
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