Understanding the Path Integral Formulation

Currently, I am reading chapter 3 of Condensed Matter Field Theory, which is on the Path Integral formulation of quantum mechanics. The book denotes $Delta t = t/N$, where $t$ is the total time considered, and $N$ is the total number of time segments. Equation 3.3 gives a Path integral approximation:

$$langle q_f |(e^{-ihat{H}Delta t/hbar})^N|q_irangle = langle q_f|I_1 e^{-ihat{T}Delta t/hbar}e^{-ihat{V}Delta t/hbar}I_2….I_Ne^{-ihat{T}Delta t/hbar}e^{-ihat{V}Delta t/hbar} |q_i rangle$$

where $I_n$ is the identity operator $$I_n=int dp_nint dq_n |q_nranglelangle q_n|p_nranglelangle p_n| .$$

I understand this part; I also understand that $$langle q_n|p_nrangle = frac{1}{sqrt{2pi hbar}}e^{ip_n q_n/hbar}.$$

However, I do not understand how the following formula is obtained. I understand that I should plug in the definitions for $I_n$ and $langle q_n|p_nrangle$, but when I do so I do not get the following. For instance I do not see why there should be a $(2pi hbar)^N$; shouldn’t there be a $(2pi hbar)^frac{N}{2}$instead? Also, where does $V(q_n)$ and $T(p_{n+1})$ come from? Also, where does the last term in the exponent come from?

$$langle q_f |(e^{-ihat{H}Delta t/hbar})^N|q_irangle = int Pi_{n=1}^{N-1} d{q_n} Pi_{n=1}^N frac{dp_n}{2pi hbar} e^{-frac{it}{hbar} sum_{n=0}^{N-1}big(V(q_n) + T(p_{n+1}) – p_{n+1}frac{q_{n+1} – q_n}{t})big)}.$$