Answer:
0.7239 +0.9435i . . . and 3 others
Step-by-step explanation:
Roots of complex numbers are most easily found when the numbers are in polar form. Then Euler's formula can be used.
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<h3>polar/exponential form</h3>
The given number, written in polar and/or exponential form is ...
z⁴ = -√3 -i = 2∠210° = 2·e^(i(7π/6+2kπ))
<h3>roots</h3>
The value of z is the fourth root of this. The fourth root of 2 multiplies the exponential term with 1/4 of the original exponent. We note there are four roots.
![\displaystyle z=\sqrt[4]{2\cdot e^{i(7\pi/6+2k\pi)}}=\sqrt[4]{2}\cdot e^{i(7\pi/24+k\pi/2)}\\\\=\begin{cases}(\sqrt[4]{2}\cos(52.5^\circ)+i\sqrt[4]{2}\sin(52.5^\circ))&\approx 0.7239+0.9435i\\(\sqrt[4]{2}\cos(142.5^\circ)+i\sqrt[4]{2}\sin(142.5^\circ))&\approx-0.9435+0.7239i\\(\sqrt[4]{2}\cos(232.5^\circ)+i\sqrt[4]{2}\sin(232.5^\circ))&\approx -0.7239-0.9435i\\(\sqrt[4]{2}\cos(322.5^\circ)+i\sqrt[4]{2}\sin(322.5^\circ))&\approx 0.9435-0.7239i\end{cases}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20z%3D%5Csqrt%5B4%5D%7B2%5Ccdot%20e%5E%7Bi%287%5Cpi%2F6%2B2k%5Cpi%29%7D%7D%3D%5Csqrt%5B4%5D%7B2%7D%5Ccdot%20e%5E%7Bi%287%5Cpi%2F24%2Bk%5Cpi%2F2%29%7D%5C%5C%5C%5C%3D%5Cbegin%7Bcases%7D%28%5Csqrt%5B4%5D%7B2%7D%5Ccos%2852.5%5E%5Ccirc%29%2Bi%5Csqrt%5B4%5D%7B2%7D%5Csin%2852.5%5E%5Ccirc%29%29%26%5Capprox%200.7239%2B0.9435i%5C%5C%28%5Csqrt%5B4%5D%7B2%7D%5Ccos%28142.5%5E%5Ccirc%29%2Bi%5Csqrt%5B4%5D%7B2%7D%5Csin%28142.5%5E%5Ccirc%29%29%26%5Capprox-0.9435%2B0.7239i%5C%5C%28%5Csqrt%5B4%5D%7B2%7D%5Ccos%28232.5%5E%5Ccirc%29%2Bi%5Csqrt%5B4%5D%7B2%7D%5Csin%28232.5%5E%5Ccirc%29%29%26%5Capprox%20-0.7239-0.9435i%5C%5C%28%5Csqrt%5B4%5D%7B2%7D%5Ccos%28322.5%5E%5Ccirc%29%2Bi%5Csqrt%5B4%5D%7B2%7D%5Csin%28322.5%5E%5Ccirc%29%29%26%5Capprox%200.9435-0.7239i%5Cend%7Bcases%7D)
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<em>Additional comment</em>
The value of z⁴ can also be written as 2∠-150°. The root found by the calculator (second attachment) has 1/4 of this negative angle, rather than 1/4 of the angle +210°.
Answer:
3/10
Step-by-step explanation:
3/5 - 3/10 = 6/10 - 3/10 = 3/10
Answer:
2x+2y+z
Step-by-step explanation:
your work for the first one is perfect!!
the second one gives you the side lengths of x+2 and x-5. the formula for the area of a rectangle is l×w so you have to multiply (x+2)(x-5) and im sure you can figure that out but the answer to that is x^2-3x-10
you set up number 3 right, you just need to continue with the distribute property which would give you 4x^2-20x+25
number 4 is the same as the other ones and the answer is 36x^2-1
number 5 is just distributing the -z^3 into everything in the parenthesis giving you -5z^3-z^4+4z^5
I cant see all your work for number 6, but I can see that your term for c is 5, when it is actually -5. it looks like you didnt subtract it to make the quadratic equal to zero and to do that you would have to subtract 5.
tbh I cant read number 7
hope that helped :)
