# How do you find all solutions to #x^4-i=0#?

4 solutions

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To find all solutions to ( x^4 - i = 0 ), we first express ( i ) in terms of its polar form, ( i = e^{i\frac{\pi}{2}} ). Now, we can rewrite the equation as ( x^4 - e^{i\frac{\pi}{2}} = 0 ). Using the De Moivre's theorem, which states that for any complex number ( z = r(\cos(\theta) + i\sin(\theta)) ), ( z^n = r^n(\cos(n\theta) + i\sin(n\theta)) ), we can find the solutions.

So, for our equation ( x^4 - e^{i\frac{\pi}{2}} = 0 ), we have ( x^4 = e^{i\frac{\pi}{2}} ). Taking the fourth root of both sides, we get:

[ x = \sqrt[4]{e^{i\frac{\pi}{2}}} ]

[ x = e^{i\frac{\pi}{8}}, \ e^{i\frac{5\pi}{8}}, \ e^{i\frac{9\pi}{8}}, \ e^{i\frac{13\pi}{8}} ]

These are the four solutions to the equation ( x^4 - i = 0 ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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