Problem 71 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

-i

Step 1: Simplify the Expression

Start by simplifying the expression \(i^{3}\). Remember that the imaginary unit \(i\) has the property that \(i^{2} = -1\). That means \(i^{3} = i^{2} \cdot i = -i\). Therefore, the given expression becomes \(\frac{1}{-i}\).

Step 2: Rationalize the Denominator

Next, rationalize the denominator by multiplying both the numerator and the denominator by the complex conjugate of -i which is i: \(\frac{1}{-i} \cdot \frac{i}{i} = -i\).

Step 3: Write in Standard Form

The complex number -i can also be written as \(0 - 1i\) which is a standard form of a complex number \(a + bi\), where \(a = 0\) and \(b = -1\).