Simplify i^27

Problem

i27

Solution

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

i1=i

i2=−1

i3=−i

i4=1

2. Divide the exponent by 4 to find the remainder, since the powers of i repeat every four terms.

27÷4=6* remainder *3

3. Rewrite the expression using the property of exponents a(m*n+r)=(an)m⋅ar

i27=i(4*(6)+3)

i27=(i4)6⋅i3

4. Substitute the known values i4=1 and i3=−i into the expression.

i27=(1)6⋅(−i)

5. Simplify the final result.

i27=1⋅(−i)

i27=−i

Final Answer

i27=−i

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