Convert to Trigonometric Form i^22+i^33+i^44+i^55

Problem

i22+i33+i44+i55

Solution

1. Simplify the powers of i by using the property that i4=1 Divide each exponent by 4 and find the remainder.

2. Evaluate each term based on the remainders: i1=i i2=−1 i3=−i and i0=1

i22=i(4*(5)+2)=i2=−1

i33=i(4*(8)+1)=i1=i

i44=i(4*(11)+0)=i0=1

i55=i(4*(13)+3)=i3=−i

3. Combine the simplified terms to find the complex number z in standard form a+b*i

z=−1+i+1−i

z=0

4. Identify the modulus and argument for the trigonometric form r*(cos(θ)+i*sin(θ)) Since z=0 the modulus r=0 The argument θ is undefined for the number zero, but it is conventionally expressed with r=0

r=√(,0+0)=0

Final Answer

i22+i33+i44+i55=0*(cos(θ)+i*sin(θ))

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