Simplify i^53

Problem

i^53

Solution

1. Identify the cycle of the powers 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. We calculate 53÷4

3. Calculate the quotient and remainder. Since 4×13=52 we have $53 = 4(13) + 1$.

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

i^53=i^(4*(13)+1)

5. Simplify using the fact that i^4=1

i^53=(i^4)^13⋅i^1

6. Substitute 1 for i^4

i^53=(1)^13⋅i

7. Evaluate the final result.

i^53=1⋅i=i

Final Answer

i^53=i

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