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I'm trying to simplify the expression

$$\left|\sqrt{a^2+ibt}\right|^2$$

where $a,b,t \in \Bbb R$.

I know that by definition

$$\left|\sqrt{a^2+ibt}\right|^2 = \sqrt{a^2+ibt}\left(\sqrt{a^2+ibt}\right)^*$$

But how do you find the complex conjugate of the square root of a complex number? And what is the square root of a complex number (with arbitrary parameters) for that matter?

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  • $\begingroup$ Hint: $|z|^2=|z^2|$. $\endgroup$ – zipirovich Sep 1 '16 at 0:33
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For any complex number $z$, and any square root $\sqrt{z}$ of $z$ (there are two), we have $$\bigl|\sqrt{z}\bigr|=\sqrt{|z|}$$ Therefore $$\bigl|\sqrt{a^2+ibt}\bigr|^2=\sqrt{|a^2+ibt|^2}=|a^2+ibt| = \sqrt{a^4+b^2t^2}$$

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  • $\begingroup$ Ah. Cool beans. Thanks! $\endgroup$ – Bobbie D Sep 1 '16 at 0:33
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In steps, when $\text{a}\space\wedge\space\text{b}\space\wedge\space\text{t}\in\mathbb{R}$:

$$\left|\sqrt{\text{a}^2+\text{b}\text{t}i}\right|^2=\left|\left(\text{a}^2+\text{b}\text{t}i\right)^{\frac{1}{2}}\right|^2=\left|\text{a}^2+\text{b}\text{t}i\right|^{2\cdot\frac{1}{2}}=\left|\text{a}^2+\text{b}\text{t}i\right|^{1}=$$ $$\left|\text{a}^2+\text{b}\text{t}i\right|=\sqrt{\Re^2\left[\text{a}^2+\text{b}\text{t}i\right]+\Im^2\left[\text{a}^2+\text{b}\text{t}i\right]}=\sqrt{\text{a}^2+\left(\text{b}\text{t}\right)^2}$$

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