The sum of all values of 'p' so that the term independent of x in the expansion of  is 45 is

To find the sum of all values of $p$ such that the term independent of $x$ in the expansion results in a constant term of 45, we first need to determine the context of the expansion.

Assuming the expansion is derived from a binomial expression, like $(x + p)^n$ or similar forms, and we need to isolate the term where the power of $x$ is zero.

Step-by-Step Solution

1. Break Down the Problem

   • Identify the expression whose expansion we are considering.
   • Define the general term in the expansion.

2. Relevant Concepts

   • For a binomial expansion $(a + b)^n$, the general term is given by: $T_k = \binom{n}{k} a^{n-k} b^k$
   • Set $T_k$ equal to the constant term (independent of $x$).

3. Analysis and Detail

   • Assuming we consider the expansion of $(x + p)^n$ where we find the term independent of $x$:
   • The term with no $x$ will occur when $k = n$ or when $x^{n-k} = x^0$.
   • Thus, $n-k = 0 \Rightarrow k = n$.

4. Calculate the Conditions

   • Setting the conditions for the independent term equal to 45: $p^n = 45$
   • Solve for $p$: $p = 45^{1/n}$
   • Depending on $n$, find the possible values of $p$.

5. Sum the Values

   • If $n$ is given, compute the roots and their sum.

Final Answer

Without loss of generality, if we only need to sum $p$ for different rational $n$ leading to integer positive roots: $\text{Sum of all values of } p = \sum_{k=1}^{n} 45^{1/k}$

Ensure you substitute any specific values for $n$ if provided to obtain the final sum. If not given, you can work on ranges of $n$ to express your findings.