Simplify i^64

Problem

i64

Solution

1. Identify the pattern of the powers of the imaginary unit i

i1=i

i2=−1

i3=−i

i4=1

2. Recognize that the powers of i repeat in a cycle of 4. Any power in can be simplified by finding the remainder of n when divided by 4.

3. Divide the exponent 64 by 4 to find the remainder.

64÷4=16

64=4×16+0

4. Apply the property of exponents to rewrite the expression using the remainder.

i64=(i4)16

5. Substitute the value i4=1 into the expression.

i64=1

6. Simplify the result.

1=1

Final Answer

i64=1

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