Answer
$\frac{P}{\rho V^{2}}=\frac{M L^{-1} T^{-2}}{\left(M L^{-3}\right)\left(L T^{-1}\right)^{2}}$ "adding and subtracting exponents".
$=(M)^{(1-1)}(L)^{(-1+3-2)}(T)^{(-2+2)}$
$=M^{0} L^{0} T^{0} \Rightarrow$ "Dimensionless"
$\frac{V x}{\nu}=\frac{\left(L T^{-1}\right)(L)}{L^{2} T^{-1}}$ "Adding and subtracting exponents".
$=(L)^{(1+1-2)}(T)^{(-1+1)}$
$=L^{0} T^{0} \Rightarrow$ "Dimensionless"
$f T=(T)^{-1}(T)$ "Adding and subtracting exponents"
$=T^{0} \Rightarrow$ 'Dimensionless"
$\frac{a T}{V}=\frac{\left(L T^{-2}\right)(T)}{L T^{-1}}$ "Adding and subtracting exponents".
$=(L)^{(1-1)}(T)^{(-2+1+1)}$
$=L^{0} T^{0} \Rightarrow$ "Dimensionless"
$\frac{a x}{\left(L T^{-1}\right)^{2}}$ "Adding and subtracting exponents".
$=(L)^{(1+1-2)}(T)^{(2-2)}$
$=L^{0} T^{0} \Rightarrow$ "Dimensionless".
Work Step by Step
$\frac{P}{\rho V^{2}}=\frac{M L^{-1} T^{-2}}{\left(M L^{-3}\right)\left(L T^{-1}\right)^{2}}$ "adding and subtracting exponents".
$=(M)^{(1-1)}(L)^{(-1+3-2)}(T)^{(-2+2)}$
$=M^{0} L^{0} T^{0} \Rightarrow$ "Dimensionless"
$\frac{V x}{\nu}=\frac{\left(L T^{-1}\right)(L)}{L^{2} T^{-1}}$ "Adding and subtracting exponents".
$=(L)^{(1+1-2)}(T)^{(-1+1)}$
$=L^{0} T^{0} \Rightarrow$ "Dimensionless"
$f T=(T)^{-1}(T)$ "Adding and subtracting exponents"
$=T^{0} \Rightarrow$ 'Dimensionless"
$\frac{a T}{V}=\frac{\left(L T^{-2}\right)(T)}{L T^{-1}}$ "Adding and subtracting exponents".
$=(L)^{(1-1)}(T)^{(-2+1+1)}$
$=L^{0} T^{0} \Rightarrow$ "Dimensionless"
$\frac{a x}{\left(L T^{-1}\right)^{2}}$ "Adding and subtracting exponents".
$=(L)^{(1+1-2)}(T)^{(2-2)}$
$=L^{0} T^{0} \Rightarrow$ "Dimensionless".
