Find root for complex number $5 + 12 i$ , $- 5 + 12$

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Solution

Methods for square root of a complex number
Let $z = \sqrt{5 + 12 i} = x + i y$ squaring
$x^{2} - y^{2} + 2 i x y = 5 + 12 i$
$x^{2} - y^{2} = 5, x y = 6 .$ solve By inspection $: 5 + 12 i$
Take half of coefficient of 'i' , i. e. $\frac{1}{2}$ (12)=6. Now decompose into two factors , squares of whose different is 5 i.e. 3 and 2 ... $3^{2} - 2^{2} = 5$
Hence $\sqrt{5 + 12 i} = \pm (3 + 2 i)$
$∴ \sqrt{5 - 12 i} = \pm (2 + 3 i) .$ conjugate
and $\sqrt{- 5 + 12 i} = \pm (2 + 3 i)$
and $\sqrt{- 5 - 12 i} = \pm (2 - 3 i)$ conjugate
Here R.P. $= - 5 = 2^{2} - 3^{2}$ . Hence we chose $2 \pm 3 i, 3 \pm 2 i$

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Similar questions

z = \frac{\sqrt{5 + 12 i} + \sqrt{5 - 12 i}}{\sqrt{5 + 12 i} - \sqrt{5 - 12 i}}

z = \frac{\sqrt{5 + 12 i} + \sqrt{5 - 12 i}}{\sqrt{5 + 12 i} - \sqrt{5 - 12 i}}

5 + 12 i

- 5 + 12 i

z_{1}, z_{2}

| z_{1} | = 20

| z_{2} - 5 - 12 i | = 4

| z_{1} - z_{2} |

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