The probability of the amount of time taken for a secretary to process a memo independent of others is modeled as an exponential random variable with PDF
$ \\ f_{T}(t) = \frac{ 1 }{ 2 }e ^{-\frac{ t }{ 2 }}$.
Also, the probability of the number of memos that the secretary is assigned daily is modeled as a Poisson RV with PMF
$$
P_{N}(k) = \frac{ L^k}{ k! }e^{-L} \mbox{ for all integers }k\ge 0
$$
where $L$ is a positive constant.
What is the total expected amount of time the secretary spends on memos per day?
Justify the claim that $N$ and $T$ are independent. If you haven't shown that for independent $X$ and $Y$, $E(XY) = E(X) E(Y)$ (this step generalizes what I imagine your argument involving sums does), do so now. Apply the result.
Edit in response to your comment:
You want to know why the value you seek is $E(NT)$.
We have: $(S|N) = T_1 + T_2 + \cdots T_N$ is the time it takes to process $N$ claims. Then
$$ E(S|N) = E(T_1) + E(T_2) + \cdots + E(T_N) = N \cdot E(T) $$
What is the expectation of that value?