Simplify the expression: [tex]i^{64} =[/tex]
Answer (2)
Recognize that i is the imaginary unit with a cyclic pattern of powers: i 1 = i , i 2 = − 1 , i 3 = − i , i 4 = 1 .
Divide the exponent 64 by 4 to find the remainder, which is 0.
Use the remainder to determine the simplified value: i 64 = i 0 = 1 .
State the final answer: 1 .
Explanation
Understanding the Problem We are asked to simplify the expression i 64 . Here, i is the imaginary unit, defined as i = − 1 . We know that i 1 = i , 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 64 , we need to find the remainder when 64 is divided by 4. This is because the powers of i repeat every 4 powers: i 1 = i , i 2 = − 1 , i 3 = − i , i 4 = 1 , i 5 = i , and so on.
Calculating the Power of i We divide 64 by 4: 64 ÷ 4 = 16 with a remainder of 0. This means that i 64 is the same as i 0 or i 4 , which is equal to 1.
Final Answer Therefore, i 64 = 1 .
Examples
Understanding powers of imaginary units is crucial in electrical engineering when analyzing alternating current circuits. For example, impedance, which is the opposition to current flow in an AC circuit, is often expressed using complex numbers involving the imaginary unit i . Simplifying expressions with powers of i helps engineers calculate and design efficient and stable circuits. This ensures that devices operate correctly and avoid damage from excessive current or voltage.
The expression i 64 simplifies to 1 because the powers of the imaginary unit i repeat every four terms. Dividing 64 by 4 gives a remainder of 0, which corresponds to i 0 = 1 . Thus, i 64 = 1 .
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