Problem

Simplify (1−i)5 and give your answer in rectangular form, polar form and exponential form.

Answer

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Answer

Final Answer: In rectangular form, the simplified expression is −4+4i. In polar form, it is 5.66∠−225∘. In exponential form, it is −4ei225∘.

Steps

Step 1 ：Express (1−i) in polar form: r=12+(−1)2=2 and θ=arctan⁡(−11)=−π4

Step 2 ：Express (1−i) in exponential form: (1−i)=2e−iπ4

Step 3 ：Raise (1−i) to the power of 5: (1−i)5=(2e−iπ4)5

Step 4 ：Simplify the expression: (1−i)5=252e−i5π4

Step 5 ：Convert back to rectangular form: (1−i)5=−4+4i

Step 6 ：Final Answer: In rectangular form, the simplified expression is −4+4i. In polar form, it is 5.66∠−225∘. In exponential form, it is −4ei225∘.

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