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Zina [86]
3 years ago
10

Use the method in the Example to find the square roots of 1 + √3i

Mathematics
1 answer:
kiruha [24]3 years ago
5 0

Answer:

\sqrt{1 + \sqrt{3} i}=\pm (1.58+i 0.548)

Step-by-step explanation:

Given

z = 1 + √3 i

Let \sqrt{1+\sqrt(3) i}=p+iq

Squaring both sides

1+\sqrt(3) i=p^2-q^2+2ipq

Comparing real and imaginary part

Re(LHS)=Re(RHS)

1=p^2-q^2...........................(1)

comparing Im(LHS)=Im(RHS)

√3=2pq

q=\frac{\sqrt{3}}{2p}

Substitute q in equation (1)

1=p^2-(\frac{\sqrt{3}}{2p})^2

p^4-p^2-0.75=0

Let x=p^2

x^2-x-0.75=0

x=\dfrac{1\pm \sqrt{1^2+4\times 0.75}}{2}

x=\frac{1\pm 4}{2}

we take only Positive value because p^2=x

x=2.5  

p^2=2.5

thus p=\pm 1.58

q=\pm 0.548

thus,

\sqrt{1 + \sqrt{3} i}=\pm (1.58+i 0.548)

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