Simplify i^46

Problem

i46

Solution

1. Identify the cycle of powers 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 46÷4

3. Determine the quotient and remainder. Since 4×11=44 we have 46 = 4(11) + 2$.

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

5. Substitute the remainder into the expression.

i46=i2

6. Evaluate the resulting power.

i2=−1

Final Answer

i46=−1

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