I found this rather strange result in the excel.

-2^-(-0.2) = -1.148698355

but

IMPOWER(-2&"+0i",-(-0.2)) = 0.9293+0.6751i

Why the different results?

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I agree: the result from IMPOWER is inexplicable. -2+0i is the same as -2. So IMPOWER should give the same result as (-2)^0.2.

Arguably, (-2)^0.2 might result in a complex number (a + bi). What do you get on a TI calculator that does arithmetic with complex numbers?

You might also be surprised by the sign of the result of -2^0.2. Excel parses that as (-2)^0.2. Mathematically, 0.2 is the rational number 2/10. So arguably, (-2)^0.2 might be evaluated as ((-2)^2)^0.1, which is +1.14869835499704, not -1.14869835499704.

However, I'm sure that Excel is not clever enough to recognize 0.2 as 2/10. And technically, it is not, since the exact decimal equivalent of the binary representation is 0.200000000000000,01110223024625156540423631668090 8203125.

(I use period for the decimal point and comma to demarcate the first 15 significant digits, which is all that Excel will format, an arbitrary limitation.)

So Excel probably evaluates (-2)^0.2 like any other exponentiation by a fractional number (e.g. 0.12345). I don't know how Excel does that. And I cannot speculate because it has been too many decades since I studied that kind of math.

In any case, I suggest that you write -(2^0.2), if that is the evaluation that you intended. Unlike the language of math, unary minus has higher precedence in Excel than exponentiation.