Simplify i^18

Problem

i^18

Solution

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

2. Divide the exponent by 4 to find the remainder.

18÷4=4* remainder *2

3. Rewrite the expression using the property of exponents i^n=i^(4*k+r)=i^r where r is the remainder.

i^18=i^(4*(4)+2)

4. Simplify the expression by substituting the known value for i^2

i^18=(i^4)^4⋅i^2

i^18=(1)^4⋅(−1)

Final Answer

i^18=−1

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