Simplify i^63

Problem

i^63

Solution

1. Identify the cycle of the powers of the imaginary unit i The values repeat every four powers: i^1=i i^2=−1 i^3=−i and i^4=1

2. Divide the exponent by 4 to find the remainder. We calculate 63÷4

3. Determine the quotient and remainder. Since 4×15=60 we have:

63=4*(15)+3

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

i^63=i^(4*(15)+3)

i^63=(i^4)^15⋅i^3

5. Substitute the known values i^4=1 and i^3=−i into the expression.

i^63=(1)^15⋅(−i)

6. Simplify the final result.

i^63=1⋅(−i)

i^63=−i

Final Answer

i^63=−i

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