Workaround? "Simplify" complicates this expression

Questions
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1

This gives the simplest answer (True):

Refine[a^b (c^d)^b == a^b c^(d b), 
 b \[Element] Integers && d \[Element] Integers]

This does not:

Simplify[a^b (c^d)^b == a^b c^(d b), 
 b \[Element] Integers && d \[Element] Integers]

My understanding of Refine vs. Simplify comes from this tutorial, which only indicates that Simplify should yield something simpler than Refine:

2

In[14]:= Simplify[a^b (c^d)^b - a^b c^(d b), b [Element] Integers && d [Element] Integers]
Out[14]= 0

In[15]:= $Version
Out[15]= "10.4.1 for Microsoft Windows (64-bit) (April 11, 2016)"
3

Thanks. That does get me around the problem.

More generally, I am curious why the algorithm used by Simplify prefers anything besides True, if anyone knows. Is there some reason for Simplify not to conform with its description in the tutorial?

4 
FullSimplify[a^b (c^d)^b == a^b c^(d b), 
 b \[Element] Integers && d \[Element] Integers]

Gives True as well (all I did was replace Simplify with FullSimplify). Generally I just use FullSimplify. It can take a little longer, but many times gets a better result.