Simplify the expressions.
[tex]
\begin{array}{l}
i^{37}=\\
i^{52}=
\end{array}
[/tex]
Answer (1)
Simplify i 37 by dividing 37 by 4, which gives a remainder of 1, so i 37 = i .
Simplify i 52 by dividing 52 by 4, which gives a remainder of 0, so i 52 = 1 .
Therefore, i 37 = i and i 52 = 1 .
The simplified expressions are i ; 1 .
Explanation
Understanding the Problem We are asked to simplify i 37 and i 52 , where i is the imaginary unit, defined as i = − 1 . We know that i 2 = − 1 , i 3 = − i , and i 4 = 1 . This pattern repeats every four powers.
Simplifying i^37 To simplify i 37 , we can divide 37 by 4 to find the remainder. 37 ÷ 4 = 9 with a remainder of 1 This means that i 37 = i 4 × 9 + 1 = ( i 4 ) 9 × i 1 = ( 1 ) 9 × i = 1 × i = i .
Simplifying i^52 To simplify i 52 , we can divide 52 by 4 to find the remainder. 52 ÷ 4 = 13 with a remainder of 0 This means that i 52 = i 4 × 13 + 0 = ( i 4 ) 13 × i 0 = ( 1 ) 13 × 1 = 1 × 1 = 1 .
Final Answer Therefore, i 37 = i and i 52 = 1 .
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 using complex numbers involving i . Simplifying expressions with i helps engineers calculate circuit behavior, design filters, and ensure stable power delivery. For example, calculating the total impedance in a series RLC circuit involves summing resistance, inductive reactance, and capacitive reactance, all potentially involving i .
