Which square root is between four and five?
1 answer:
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By using <span>De Moivre's theorem:
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If we have the complex number ⇒ z = a ( cos θ + i sin θ)
∴![\sqrt[n]{z} = \sqrt[n]{a} \ (cos \ \frac{\theta + 360K}{n} + i \ sin \ \frac{\theta +360k}{n} )](https://tex.z-dn.net/?f=%20%5Csqrt%5Bn%5D%7Bz%7D%20%3D%20%20%5Csqrt%5Bn%5D%7Ba%7D%20%5C%20%28cos%20%5C%20%20%5Cfrac%7B%5Ctheta%20%2B%20360K%7D%7Bn%7D%20%2B%20i%20%5C%20sin%20%5C%20%5Cfrac%7B%5Ctheta%20%2B360k%7D%7Bn%7D%20%29)
k= 0, 1 , 2, ..... , (n-1)
For The given complex number <span>⇒ z = 81(cos(3π/8) + i sin(3π/8))
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Part (A) <span>find the modulus for all of the fourth roots
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<span>∴ The modulus of the given complex number = l z l = 81
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∴ The modulus of the fourth root =![\sqrt[4]{z} = \sqrt[4]{81} = 3](https://tex.z-dn.net/?f=%20%5Csqrt%5B4%5D%7Bz%7D%20%3D%20%20%5Csqrt%5B4%5D%7B81%7D%20%3D%203)
Part (b) find the angle for each of the four roots
The angle of the given complex number =
There is four roots and the angle between each root =
The angle of the first root =
The angle of the second root =
The angle of the third root =
The angle of the fourth root =
Part (C): find all of the fourth roots of this
The first root =
The second root =
The third root =
The fourth root =
</span>
If we have the complex number ⇒ z = a ( cos θ + i sin θ)
∴
k= 0, 1 , 2, ..... , (n-1)
For The given complex number <span>⇒ z = 81(cos(3π/8) + i sin(3π/8))
</span>
Part (A) <span>find the modulus for all of the fourth roots
</span>
<span>∴ The modulus of the given complex number = l z l = 81
</span>
∴ The modulus of the fourth root =
Part (b) find the angle for each of the four roots
The angle of the given complex number =
There is four roots and the angle between each root =
The angle of the first root =
The angle of the second root =
The angle of the third root =
The angle of the fourth root =
Part (C): find all of the fourth roots of this
The first root =
The second root =
The third root =
The fourth root =