Simplifying Powers of $i$
Use the inverse relationship to complete the expression.
If $i=\sqrt{-1}$, then $i^2=-1$.
Using laws of exponents, $i^3=\left(i^2\right) i=$ $\square$
Answer (1)
Express i 3 as i 2 i .
Substitute i 2 = − 1 into the expression.
Simplify to get i 3 = − i .
Therefore, i 3 = − i .
Explanation
Understanding the Problem We are given that i = − 1 and i 2 = − 1 . We want to simplify i 3 using the laws of exponents.
Expressing i 3 We can express i 3 as a product of i 2 and i using the laws of exponents: i 3 = i 2 ⋅ i .
Substituting i 2 Now, we substitute the value of i 2 = − 1 into the expression: i 3 = ( − 1 ) ⋅ i .
Simplifying Simplifying the expression, we get i 3 = − i .
Examples
Complex numbers, including powers of i , are used in electrical engineering to analyze alternating current circuits. The imaginary unit i helps represent the phase difference between voltage and current. For example, the impedance of a circuit can be expressed as a complex number involving i , allowing engineers to calculate current and voltage accurately. This is crucial for designing efficient and stable electrical systems.
