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

Question

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

Accepted Answer

∻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.⚹◊What is the formula for determining the square roots of a complex number in the form $a+bi$?⍭ Generate me a similar question◊

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