Given:
[tex]$\frac{\left(\frac{a^{2}-b^{2}}{a b}\right)}{\left(\frac{1}{b}-\frac{1}{a}\right)}[/tex]
To find:
The simplified expression.
Solution:
[tex]$\frac{\left(\frac{a^{2}-b^{2}}{a b}\right)}{\left(\frac{1}{b}-\frac{1}{a}\right)}[/tex]
Apply the fraction rule: [tex]\frac{\frac{y}{z}}{x}=\frac{y}{z \cdot x}[/tex]
[tex]$\frac{\left(\frac{a^{2}-b^{2}}{a b}\right)}{\left(\frac{1}{b}-\frac{1}{a}\right)}=\frac{a^{2}-b^{2}}{a b\left(\frac{1}{b}-\frac{1}{a}\right)}[/tex] ------------ (1)
Let us simplify [tex]\frac{1}{b}-\frac{1}{a}[/tex].
LCM of a and b = ab
Adjacent fraction based on the LCM
[tex]\frac{1}{b}-\frac{1}{a}=\frac{a}{a b}-\frac{b}{a b}[/tex]
[tex]=\frac{a-b}{a b}[/tex]
Substitute this in equation (1).
[tex]$=\frac{a^{2}-b^{2}}{\frac{a-b}{a b} a b}[/tex]
Common factor ab get canceled.
[tex]$=\frac{a^{2}-b^{2}}{a-b}[/tex]
Apply the algebraic formula: [tex]x^{2}-y^{2}=(x+y)(x-y)[/tex]
[tex]$=\frac{(a+b)(a-b)}{a-b}[/tex]
Cancel the common factor a - b, we get
[tex]=a+b[/tex]
The simplified expression is a + b.
