Regroup terms.
\[z-{a}^{3}azbb={a}^{3}a-b\]
Use Product Rule: \({x}^{a}{x}^{b}={x}^{a+b}\).
\[z-{a}^{3+1}z{b}^{1+1}={a}^{3}a-b\]
Simplify \(3+1\) to \(4\).
\[z-{a}^{4}z{b}^{1+1}={a}^{3}a-b\]
Simplify \(1+1\) to \(2\).
\[z-{a}^{4}z{b}^{2}={a}^{3}a-b\]
Simplify \({a}^{3}a\) to \({a}^{4}\).
\[z-{a}^{4}z{b}^{2}={a}^{4}-b\]
Factor out the common term \(z\).
\[z(1-{a}^{4}{b}^{2})={a}^{4}-b\]
Rewrite \(1-{a}^{4}{b}^{2}\) in the form \({a}^{2}-{b}^{2}\), where \(a=1\) and \(b={a}^{2}b\).
\[z({1}^{2}-{({a}^{2}b)}^{2})={a}^{4}-b\]
Use Difference of Squares: \({a}^{2}-{b}^{2}=(a+b)(a-b)\).
\[z(1+{a}^{2}b)(1-{a}^{2}b)={a}^{4}-b\]
Divide both sides by \(1+{a}^{2}b\).
\[z(1-{a}^{2}b)=\frac{{a}^{4}-b}{1+{a}^{2}b}\]
Divide both sides by \(1-{a}^{2}b\).
\[z=\frac{\frac{{a}^{4}-b}{1+{a}^{2}b}}{1-{a}^{2}b}\]
Simplify \(\frac{\frac{{a}^{4}-b}{1+{a}^{2}b}}{1-{a}^{2}b}\) to \(\frac{{a}^{4}-b}{(1+{a}^{2}b)(1-{a}^{2}b)}\).
\[z=\frac{{a}^{4}-b}{(1+{a}^{2}b)(1-{a}^{2}b)}\]
z=(a^4-b)/((1+a^2*b)*(1-a^2*b))
