Perform the indicated operations, and write the result in the form $a+b i$. (Simplify your answers completely.)
$i^{44}$
Answer (1)
We are asked to evaluate i 44 .
Divide the exponent 44 by 4 to find the remainder, which is 0.
Since the remainder is 0, i 44 = i 0 = 1 .
Write 1 in the form a + bi , which is 1 + 0 i .
Explanation
Understanding the Problem We are asked to evaluate i 44 and express it in the form a + bi , where a and b are real numbers. The imaginary unit i is defined as i = − 1 , so i 2 = − 1 , i 3 = − i , and i 4 = 1 . The powers of i repeat in a cycle of 4: i 1 = i , i 2 = − 1 , i 3 = − i , i 4 = 1 , i 5 = i , and so on. To find i 44 , we need to determine where 44 falls in this cycle.
Finding the Remainder To find the equivalent power of i , we divide the exponent 44 by 4 and find the remainder. The remainder will tell us which value in the cycle i , − 1 , − i , 1 corresponds to i 44 .
Calculating the Power When we divide 44 by 4, we get 44 ÷ 4 = 11 with a remainder of 0. This means that i 44 is equivalent to i 0 , which is equal to 1.
Expressing in a+bi Form Now we write 1 in the form a + bi . Since 1 is a real number, we can write it as 1 + 0 i . Therefore, i 44 = 1 + 0 i .
Final Answer Thus, the result of the operation is 1 + 0 i .
Examples
Understanding powers of i is crucial in electrical engineering when analyzing alternating current (AC) circuits. Impedance, which is the AC equivalent of resistance, is often expressed as a complex number involving i . Calculating powers of i helps simplify complex circuit equations and determine the behavior of AC circuits. For example, analyzing the stability of a circuit might involve evaluating expressions with i 44 to ensure the circuit operates as expected.
