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Chapter 2: Problem 72

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

Short Answer

Expert verified
The complex number simplified into standard form is \(0 - \frac{1}{8}i\)

Step by step solution

01

Simplify the denominator

Start by simplifying the denominator as follows: \((2 i)^{3} = 8i^{3}\). Now, remember that \(i = \sqrt{-1}\), hence \(i^{2} = -1\) and \(i^{3} = -i\). Thus, the denominator simplifies to \(-8i\).
02

Simplify the fraction

To simplify the fraction, multiply the numerator and denominator by the complex conjugate of the denominator. The complex conjugate of \(-8i\) is \(8i\). This will help get rid of 'i' from the denominator. The simplified form becomes \(\frac{1 \cdot 8i}{-8i \cdot 8i} = \frac{8i}{-64} = -\frac{8}{64}i\).
03

Convert to standard form

Step 2 simplified the complex number to be in form of a real number times 'i'. This is \(a + bi\) where \(a = 0\) and \(b = -\frac{8}{64}\), simplifying b we get \(b = -\frac{1}{8}\). Thus, our complex number in standard form is \(0 - \frac{1}{8}i\)

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Most popular questions from this chapter

Solving an Equation Involving Fractions Find all solutions of the equation. Check your solutions. $$x=\frac{3}{x}+\frac{1}{2}$$

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