Problem 72 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \(-8i\).

Step 2: 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 \(\frac{1 \cdot 8i}{-8i \cdot 8i} = \frac{8i}{-64} = -\frac{8}{64}i\).

Step 3: 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 \(0 - \frac{1}{8}i\)