# Unexpected behavior of Exponent

Fixed in Version 14.0

According to Exponent:

• Exponent[expr, form] gives the maximum power with which form appears in the expanded form of expr.
• Exponent[expr, form, h] applies h to the set of exponents with which form appears in expr.
• The default taken for h is Max.

So Exponent[..., …, Max] is equivalent to Exponent[..., …]. But how to explain the following output?

In[1]:= Exponent[x, Sqrt[x]]

Out[1]= 2

In[2]:= Exponent[x, Sqrt[x], Max]

Out[2]= 1

In[3]:= Exponent[Sqrt[x], Sqrt[x], Defer]

1
Out[3]= Defer[-]
2



Isn't this a bug?

• It seems weird, indeed. For example, you can use List as the 3rd argument to get all the exponents. But Exponent[x, Sqrt[x]] gives 2 and Exponent[x, Sqrt[x], List] gives {1}. Sep 12, 2023 at 7:40
• The documentation says under Possible Issues that Exponent is purely syntactical and does no analysis. Sep 12, 2023 at 8:51
• @RolandF If so, shouldn't Exponent[x, Sqrt[x]] return 0 (instead of 2)? And shouldn't Exponent[Sqrt[x], Sqrt[x], Identity] return 1 (rather than 1/2)? Sep 12, 2023 at 10:21
• By experiment, Exponent[x^a, x^b] works reasonably with numbers as exponents of symbols, but not with exponent symbols. Even specifying everything] as positive integers or expecting simplifications as in Exponent[(1-x^2)/(1+x),x] returns input. Using a non-symbol expression as variable in polynomial reduction may be termed 'abuse of powers'. Sep 12, 2023 at 10:58

Fixed in Version 14.0, now Exponent[Sqrt[x], Sqrt[x], Defer] returns 1/2