Find All Complex Number Solutions z=-4 square root of 3-4i

Problem

z=−4√(,3)−4*i

Solution

1. Identify the real and imaginary parts of the complex number z=a+b*i

a=−4√(,3)

b=−4

2. Calculate the modulus r using the formula r=√(,a2+b2)

r=√(,(−4√(,3))2+(−4)2)

r=√(,48+16)

r=√(,64)=8

3. Determine the argument θ using tan(θ)=b/a

tan(θ)=(−4)/(−4√(,3))=1/√(,3)

4. Locate the quadrant of the complex number. Since both a and b are negative, the number lies in the third quadrant.

θ=π+π/6=(7*π)/6

5. Write the complex number in polar form z=r*(cos(θ)+i*sin(θ))

z=8*(cos((7*π)/6)+i*sin((7*π)/6))

6. Express the solution in exponential form z=r*e(i*θ)

z=8*e(i(7*π)/6)

Final Answer

z=8*(cos((7*π)/6)+i*sin((7*π)/6))

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