Given:
[tex]$\frac{a-a b}{a^{2}} \div \frac{a-1}{a^{3}}[/tex]
Solution:
[tex]$\frac{a-a b}{a^{2}} \div \frac{a-1}{a^{3}}[/tex]
Apply the fraction rule:
[tex]$\frac{w}{x} \div \frac{y}{z}=\frac{w}{x} \times \frac{z}{y}[/tex]
Using this rule,
[tex]$\frac{a-a b}{a^{2}} \div \frac{a-1}{a^{3}}=\frac{a-a b}{a^{2}} \times \frac{a^{3}}{a-1}[/tex]
Factor out common term a in first fraction.
[tex]$=\frac{a(1-b)}{a^{2}}\times \frac{a^{3}}{a-1}[/tex]
Cancel the common factor a.
[tex]$=\frac{1-b}{a} \times \frac{a^{3}}{a-1}[/tex]
Apply the multiplication of fraction rule:
[tex]$\frac{w}{x} \times \frac{y}{z}=\frac{w \times y}{x \times z}[/tex]
Using this rule, we get
[tex]$=\frac{(1-b) a^{3}}{a(a-1)}[/tex]
Cancel the common factor a.
[tex]$=\frac{a^{2}(1-b)}{a-1}[/tex]
Therefore,
[tex]$\frac{a-a b}{a^{2}} \div \frac{a-1}{a^{3}}=\frac{a^{2}(1-b)}{a-1}[/tex]
