Elissa89 said:So I have a study guide for my final which was written by a different professor from my actual professor. So I don't understand the question, I don't know if it's because my professor did not teach this or if the wording is different from what I'm used to:
Find the square roots of 4*sqrt(3)+4(i)
MarkFL said:I would let:
\(\displaystyle y=8\left(\frac{\sqrt{3}}{2}+\frac{1}{2}i\right)=8e^{\Large\frac{\pi}{6}i}\)
Can you proceed?
Actually I don' know what any of that means. Where did the 8 come from?Elissa89 said:No, i don't know what the right side means.
Elissa89 said:No, i don't know what the right side means.
- - - Updated - - -Actually I don' know what any of that means. Where did the 8 come from?
The expression 4*sqrt(3)+4(i) is a complex number, which can be written in the form a+bi, where a and b are real numbers and i is the imaginary unit. In this case, a=4*sqrt(3) and b=4. Therefore, the value of the expression is 4*sqrt(3)+4i.
The square root of a complex number is calculated by using the formula z = sqrt(r)*[cos(θ/2) + i*sin(θ/2)], where z is the complex number, r is the modulus or absolute value of the complex number, and θ is the argument or angle of the complex number. In this case, the modulus is 4 and the argument is 60 degrees (since 4*sqrt(3) can be written as 4*sqrt(3)*cos(60) + 4*sqrt(3)*sin(60)i). Therefore, the square root of 4*sqrt(3)+4(i) is 2*sqrt(2)*[cos(30) + i*sin(30)].
Yes, the square root of a complex number can have multiple values. This is because the square root function is not a one-to-one function for complex numbers. In the case of 4*sqrt(3)+4(i), the square root can also be written as -2*sqrt(2)*[cos(30) + i*sin(30)].
The square root of a complex number can be represented graphically by plotting the complex number on the complex plane and then drawing a line from the origin to the point representing the square root of the complex number. In the case of 4*sqrt(3)+4(i), the square root can be represented by a point at a distance of 2*sqrt(2) from the origin and at an angle of 30 degrees from the positive real axis.
The square root of a complex number can be verified by squaring the value and checking if it equals the original complex number. In the case of 4*sqrt(3)+4(i), squaring the value of 2*sqrt(2)*[cos(30) + i*sin(30)] gives us 4*sqrt(3)+4(i), thus verifying that it is the square root of the original complex number.