[tex]$\frac{(x+1)(x-3)}{x^{2}}[/tex]
Solution:
Given expression:
[tex]$\frac{\frac{36}{x^{2}}+\frac{36}{x}}{\frac{36}{x-3}}[/tex]
To solve this expression:
[tex]$\frac{\frac{36}{x^{2}}+\frac{36}{x}}{\frac{36}{x-3}}[/tex]
Apply the fraction rule: [tex]$\frac{a}{\frac{b}{c}}=\frac{a \cdot c}{b}[/tex]
[tex]$=\frac{\left(\frac{36}{x^{2}}+\frac{36}{x}\right)(x-3)}{36}[/tex]
Let us solve [tex]\frac{36}{x^{2}}+\frac{36}{x}[/tex].
Least common multiple of [tex]x^{2}, x[/tex] is [tex]x^{2}[/tex].
Make the denominator same based on the LCM.
So that multiply and divide the 2nd term by x, we get
[tex]$\frac{36}{x^{2}}+\frac{36}{x}=\frac{36}{x^{2}}+\frac{36 x}{x^{2}}[/tex]
[tex]$=\frac{36+36 x}{x^{2}}[/tex]
Now, multiply by (x - 3).
[tex]$\frac{36+36 x}{x^{2}}(x-3)= \frac{(36 x+36)(x-3)}{x^{2}}[/tex]
[tex]$\frac{\left(\frac{36}{x^{2}}+\frac{36}{x}\right)(x-3)}{36}=\frac{\frac{(36 x+36)(x-3)}{x^{2}}}{36}[/tex]
Apply the fraction rule: [tex]\frac{\frac{b}{c}}{a}=\frac{b}{c \cdot a}[/tex]
[tex]$=\frac{(36x+36)(x-3)}{x^{2} \cdot 36}[/tex]
[tex]$=\frac{36(x+1)(x-3)}{x^{2} \cdot 36}[/tex]
Cancel the common factor 36.
[tex]$=\frac{(x+1)(x-3)}{x^{2}}[/tex]
Hence the solution is [tex]\frac{(x+1)(x-3)}{x^{2}}[/tex].
