Write in Standard Form (1-i)^10

Problem

(1−i)10

Solution

1. Identify the complex number z=1−i and convert it to polar form r*(cos(θ)+i*sin(θ))

2. Calculate the modulus r=√(,1+(−1)2)=√(,2)

3. Determine the argument θ by finding arctan((−1)/1) which gives θ=−π/4

4. Apply De Moivre's Theorem, which states that zn=rn*(cos(n*θ)+i*sin(n*θ))

5. Substitute the values into the formula: (√(,2))10*(cos(10⋅−π/4)+i*sin(10⋅−π/4))

6. Simplify the exponent and the angles: 2*(cos(−(5*π)/2)+i*sin(−(5*π)/2))

7. Evaluate the trigonometric functions using coterminal angles: −(5*π)/2 is coterminal with −π/2

8. Calculate the final values: cos(−π/2)=0 and sin(−π/2)=−1

9. Distribute the constant: 32(0 - i) = -32i$.

Final Answer

(1−i)10=−32*i

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