Simplify the following exponential algebraic expression?

Questions
1

I want to simplify an expression,

((1 + beta1 E^(d - r1 t))^(-1/beta1))^beta1 // Simplify // Normal

But it didn’t work.
Would you like to help me?
Thanks.

QQ图片20180504202901.png

2

What do you expect the result to be? Do you implicitly make any further assumptions?

Simplify[((1 + beta1 E^(d - r1 t))^(-1/beta1))^beta1, Assumptions -> beta1 > 1]
(*1/(1 + beta1 E^(d - r1 t))*)

Cheers,

Marco

3

Marco,
Your way works. But if beta1>0?

In[22]:= eq5 = (D[X1[t], t]/X1[t] - 
       r1/beta1*(1 - (X1[t]/K1)^
           beta1)) /. {X1 -> ((K1*(1 + beta1*Exp[d - r1*#1])^(-1/
              beta1)) &)} // Factor // FullSimplify // Normal

Out[22]= (E^d r1)/(
 beta1 E^d + E^(
  r1 t)) + ((-1 + ((1 + beta1 E^(d - r1 t))^(-1/beta1))^
    beta1) r1)/beta1

In[44]:= eq6 = Simplify[eq5, beta1 > 1]

Out[44]= 0

QQ图片20180505110841.png
QQ图片20180505110841.png686×211 5.87 KB

4

The problem is when 1 + beta1 E^(d - r1 t) < 0, because if you “simplify” the powers the value changes:

values = {d -> r1 t + Log[4], beta1 -> -1/2};
eq5 /. values
PowerExpand[eq5] /. values

It is not always true that (a^b)^(1/b) is the same as a.