answer should be plus minus (3+i root2)

`find the complex square roots of 7+6root21 giving your answer in the fore x+iy, where x and y are real`

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Answer from Sia

Posted 10 days ago

Solution by Steps

step 1

To find the complex square roots of $7 + 6\sqrt{21}$, we start by expressing the number in polar form

step 2

The modulus of $7 + 6\sqrt{21}$ is given by $\sqrt{(7)^2 + (6\sqrt{21})^2} = \sqrt{49 + 756} = \sqrt{805}$

step 3

The argument of $7 + 6\sqrt{21}$ is $\theta = \tan^{-1}\left(\frac{6\sqrt{21}}{7}\right)$

step 4

The principal square root is given by $\sqrt{7 + 6\sqrt{21}} = \sqrt{\sqrt{805}} e^{i \frac{\theta}{2}}$

step 5

The second square root is given by $\sqrt{7 + 6\sqrt{21}} e^{i \left(\frac{\theta}{2} + \pi\right)}$

step 6

Using the Asksia-LL calculator result, the principal root is approximately $5.8733$ (real part)

step 7

The second root is approximately $-5.8733$ (real part)

Answer

The complex square roots of $7 + 6\sqrt{21}$ are approximately $5.8733$ and $-5.8733$.

Key Concept

Complex Square Roots

Explanation

To find the complex square roots of a number, we express the number in polar form and then use the properties of exponents and roots in the complex plane.

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