How to simplify polynomial at my Taylor series ?

Questions
1

The following formula u^6 Sqrt[u^3 ], how to simplify to
u^(15/2)

1:

123.png
123.png1719Ɨ709 50.9 KB

123.png
123.png1719Ɨ709 50.9 KB

taylor.nb (11.2 KB)

2

t1 = u^6 Sqrt[u^3]
{t1, (1/t1)} /. {
Power[Power[u_, n_], Rational[p_, m_]] ā†’ Power[u, (p n)/m]}

3

Hi, Michael

Thank you for your code . How can I show the function like that

1234.png
1234.png1920Ɨ1080 86.5 KB

4

Hope this helps. I used E2 etc instead of E_underscore_2

t2 = carlson[E2, 
     E3] /. {Power[Power[u_, n_], Rational[p_, m_]] -> 
      Power[u, ( p n)/m]} // Normal // Collect[#, {u}] &
5

Hello

Thank you for your code again.
How to put

u^(3/2) at first item ,

u^(7/2) at 2nd item,

ā€¦

u^(23/2) at last item,

6

I am not sure that I understand exactly what you are intended to do but you can pull the polynomial expression apart. First you temporarily replace the u by a variable v with positive integer exponents:

t3 = t2 /. Power[ u_, Rational[p_, m_]] -> Power[v, -(p/m + 1/2)]

Now you use CoefficientList to extract the coefficients:

t4 = CoefficientList[t3, {v}];

Finally you bring back u into a separate list. The result looks like

t5 = {v^Range[0, Length[t4] - 1] /. {Power[v, n_] -> 
     Power[u, -n - 1/2], v -> u^(-3/2)}, t4}

Here is a check that this decomposition corresponds in fact to T2

Total[Map[Apply[Times, #] &, t5\[Transpose]]] == t2

Hope that helps

7

Michael, thank you again. You had shooting my all problem. How can I study those setting ? Do you have web sit or book else.