Simplify i^57

Problem

i57

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 57÷4

3. Calculate the quotient and remainder. Since 56 is the largest multiple of 4 less than 57 we have 57=4×14+1

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

i57=i(4×14+1)

5. Simplify using the fact that i4=1

i57=(i4)14×i1

6. Substitute 1 for i4

i57=1×i

7. Evaluate the final result.

i57=1×i

Final Answer

i57=i

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