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Use the polar representation to write the following in the form a + bi (a) (-1+ i)^7 (b) (1+v3i)^10

User Anabar
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1 Answer

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Answer with Step-by-step explanation:

For any complex number x+iy the polar form is represented as


z=re^(i\theta )

where


r^(2)=x^2+y^2\\\\tan(\theta )=(y)/(x)

Part a) x+iy = -1+i

Thus
r=√(-1^2+1^2)=√(2)\\\\  \theta _(1)=tan^(-1)((-1)/(1))=(-\pi )/(4)

Thus using De-Morvier's theorem


(-1+i)^7=(√(2)e^{i(-\pi )/(4)})^7\\\\\therefore (-1+i)^7=2^(7/2)\cdot e^{(i-\pi )/(28)}

Part 2) x+iy = 1+3i

Thus
r=√(1^2+3^2)=√(10)\\\\  \theta _(1)=tan^(-1)((3)/(1))=1.25

Thus using De-Morvier's theorem


(1+3i)^(10)=(√(10)e^(1.25i))^(10)\\\\\therefore (1+3i)^(10)=10^(5)\cdot e^(12.5i)

User Jim Wrubel
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