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OleMash [197]
3 years ago
10

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

Mathematics
1 answer:
ira [324]3 years ago
3 0

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 )=\frac{y}{x}

Part a) x+iy = -1+i

Thus r=\sqrt{-1^2+1^2}=\sqrt{2}\\\\  \theta _{1}=tan^{-1}(\frac{-1}{1})=\frac{-\pi }{4}

Thus using De-Morvier's theorem

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

Part 2) x+iy = 1+3i

Thus r=\sqrt{1^2+3^2}=\sqrt{10}\\\\  \theta _{1}=tan^{-1}(\frac{3}{1})=1.25

Thus using De-Morvier's theorem

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

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