Hot Question on Cool Topic of Power #2
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1 solution
Wrong. Where's your parentheses? You're wrong in the first line itself. You failed to justify why it's a non real number from 3rd step to 4th step.
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Can you provide a solution for this problem? After the (-1)^\sqrt[81]{-2} , I didn't manage to go any further.
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Show that the (-2)^(1/81) is an irrational number.
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@Pi Han Goh – Sure, it is irrational, but since its (-1) that we are dealing with, shouldn't the only possible answers be 1 or -1? The problem I have it's that it isn't possible to define if 2^1/81 is even or odd. So, as far I can see, the answer would be a real number, but an undefined one.
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@Nowras Otmen – Whoops. I meant to say that we should prove that it's not a real number. With (-2)^(1/81) being a complex number, then so is (-1)^( (-2)^(1/81) ).
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@Pi Han Goh – Sorry to be bothering you again, but isn't 8 1 − 2 a real number? Since 81 is odd, it shouldn't matter if the radicand is positive or negative, it will be real nonetheless, right? (e.g. 3 − 8 ) Also, I have yet to study complex numbers in depth, so I apologize in case I commit any mistakes regarding that subject.
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@Nowras Otmen – Not it's not real. y = x^(1/81) is only defined (real) for x > 0. so with -2 < 0, then (-2)^(1/81) is not real.
But -2^1/81. Could be a negative yet a very small number. For this whole expression to be unreal no. the result obtained in -2^... should be a no. with 1 as the numerator and an odd no. as the denominator.
− 1 − 2 − 3 − 4 = − 1 − 2 ( − 3 4 1 ) = − 1 − 2 ( 8 1 1 ) ⟹ Not a real no.