(-81)^\frac{1}{4}
Firstly there is no real solution for this because no real number to the power of 4 is going to be negative.
But I will give you the complex solution :)
\((-81)^\frac{1}{4}\\ =(-1)^\frac{1}{4}*(81)^\frac{1}{4}\\ =(-1)^\frac{1}{4}*3\\ =((-1)^\frac{1}{2})^\frac{1}{2}*3\\ =i^\frac{1}{2}*3\)
Now
\(\sqrt{i}=\sqrt{\frac{1+2i-1}{2}}=\sqrt{\frac{1+2i+i^2}{2}}=\sqrt{\frac{(1+i)^2}{2}}=\frac{1+i}{\sqrt2}\\ \)
\(=3*\sqrt{i}\\ =\frac{3(1+i)}{\sqrt2}\\ =\frac{3\sqrt2 (1+i)}{2}\\\)
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