Simplify i^65

Problem

i65

Solution

1. Identify the cycle of the imaginary unit i The powers of i repeat every four terms: i1=i i2=−1 i3=−i and i4=1

2. Divide the exponent by 4 to find the remainder. We calculate 65÷4

3. Calculate the quotient and remainder. Since 65=4×16+1 the remainder is 1

4. Apply the property in=ir where r is the remainder of n÷4

i65=i(4*(16)+1)

5. Simplify the expression using the laws of exponents.

i65=(i4)16×i1

6. Substitute i4=1 into the expression.

i65=1×i

7. Evaluate the final result.

i65=1×i

Final Answer

i65=i

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