Simplify the expression. [tex]i^{37}=[/tex]
Answer (1)
Recognize that the powers of i repeat every 4 powers.
Divide the exponent 37 by 4 to find the remainder, which is 1.
Use the remainder to simplify the expression: i 37 = i 4 × 9 + 1 = ( i 4 ) 9 × i 1 = i .
The simplified form of i 37 is i .
Explanation
Understanding the Problem We need to simplify i 37 . Recall that i = − 1 , so i 2 = − 1 , i 3 = − i , and i 4 = 1 . The powers of i repeat in a cycle of 4.
Finding the Remainder To simplify i 37 , we divide the exponent 37 by 4 to find the remainder. We have 37 = 4 × 9 + 1 .
Simplifying the Expression Then, we can write i 37 = i 4 × 9 + 1 = ( i 4 ) 9 × i 1 = 1 9 × i = 1 × i = i .
Final Answer Therefore, i 37 = 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 represented using complex numbers. Simplifying powers of i helps engineers calculate impedance and analyze circuit behavior, ensuring efficient and stable electrical systems. For example, in circuit analysis, i represents a 90-degree phase shift, and calculating i 37 helps determine the overall phase shift in a complex circuit.
