So I plotted the graph of y=x^(1/n) where n was varied and I notice that for some values of n like 4.4 , -0.8 left arm of graph was not there I.e. no values of y for negative x . I didn’t understand this . Please explain .
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In general, negative numbers to non-integer exponents are not automatically defined, for example, what do you do with $-1$ to the $1/2$ power, which would be the square root of $-1$? Yes, a few rational exponents like $1/3$ make sense with negative bases, but a typical graphing program simply ignores negative numbers to real non-integer powers because there is no coherent way to define them as real values (other than rational exponents with odd denominators such as $1/3$) – Ned Jan 25 '20 at 13:55
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What would be the general value of n for which we can’t take negative values of x . Like 4.4 contains both 2 , 11, 5 so ? – yohan kumar Jan 25 '20 at 16:07
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In theory, the exponent would need to be rational with an odd denominator in order to work on a negative base, which means if your exponent is $1/n$ then $n$ would have to be rational with odd numerator -- and you'd have to insist that the rational be written in lowest terms. I doubt that typical graphing programs would check for that, since they probably work with floating point (i.e. decimal) representations rather than exact (rational) numbers -- they would simply not evaluate negative bases to non-integer powers in the context of graphing. – Ned Jan 26 '20 at 00:24