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Chapter 2: Problem 88

Simplify the complex number and write it in standard form. $$\frac{1}{(2 i)^{3}}$$

Short Answer

Expert verified
The simplified form of the given complex number is \(-0.125i\).

Step by step solution

01

Simplify the denominator

Firstly, the expression in the denominator needs to be simplified. This is accomplished by finding the cube of \(2i\), which would be \(8i^3\). However, since \(i^2 = -1\), we get \(8i^3 = -8i\).
02

Compute the reciprocal

Next, the task is to compute the reciprocal of the complex number. In this case, taking the reciprocal (or multiplicative inverse) of \(-8i\) results in \(-\frac{1}{8i}\).
03

Convert to Standard form

The last step is to convert our obtained complex number into its standard form. The standard form of a complex number is \(a + bi\), where \(a\) and \(b\) are real numbers. Our current number, \(-\frac{1}{8i}\), is not in this form. Multiplying top and bottom by the complex conjugate (which is \(-i\)), we get \( \frac{(-1)(-i)}{(8i)(-i)} \). Using the fact that \(i^2 = -1\) again, we get \(\frac{i}{-8} = \frac{-1}{8}i\).
04

Final Simplification

The final simplification is done to get rid of the fraction and put the answer in simpler terms. Therefore, we have \(-0.125i\) as the final, simplified form of the complex number in question.

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Most popular questions from this chapter

Think About It Let \(y=f(x)\) be a cubic polynomial with leading coefficient \(a=-1\) and \(f(2)=f(i)=0\) Write an equation for \(f\)

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