Evaluating a simple complex expression

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NewtonianAlch
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Homework Statement



ℝ(3-7i)^4

Is this asking for just the real part of this complex expression? I suppose I could go about expanding the whole thing out, but is there a more convenient method? We've learned a few methods, such as expressing it in polar form and such. But here the angle would be tan inverse of (-7/3) which renders it to some -66.80140949...

which makes it pretty hard to express in nice Pi angles.
 
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I think expanding is the simplest. First, expand (3-7i)^2 and simplify, then square again.

ehild
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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