Answer: \(a=\frac{10}{3}\), \(b=\frac{7}{3}\), \(c=\frac92\); \(p=-2.32\) or \(p=0.585\).

(a) Write each factor using powers:

Also,

Dividing by \(x^{-3}z^{-5/2}\) is the same as multiplying by \(x^3z^{5/2}\). Hence

Therefore \(a=\frac{10}{3}\), \(b=\frac{7}{3}\) and \(c=\frac92\).

(b) Let \(u=2^p\). Since \(2^{2p+1}=2(2^p)^2=2u^2\), the equation becomes

\(10u^2-17u+3=0.\)

Factorising,

\((5u-1)(2u-3)=0.\)

So \(u=\frac15\) or \(u=\frac32\). Therefore

\(2^p=\frac15\quad\text{or}\quad 2^p=\frac32.\)

Taking logarithms gives

Hence

\(p=-2.32\quad\text{or}\quad p=0.585\)

to 3 significant figures.