Respuesta :
The least common denominator of a rational expression is [tex]6x^{2} (x+2)[/tex].
Least common denominator
Given:
[tex]5x^{2} -\frac{3}{6x^{2} +12x}[/tex]
First, find the factors for each of the denominators individually:
[tex]&x^{2}=x \cdot x \\[/tex]
[tex]&6 x^{2}+12 x=6 \cdot x \cdot(x+2)[/tex]
The common factor is: [tex]x[/tex]
Removing this leaves the following factors from each of the terms:
[tex]$x$[/tex] and [tex]$6 \cdot(x+2)$[/tex]
We need to multiply the fraction on the left by [tex]$6(x+2)$[/tex] to obtain a common denominator:
[tex]\frac{6(x+2)}{6(x+2)} \times \frac{5}{x^{2}}[/tex]
[tex]\Rightarrow \frac{5 \cdot 6(x+2)}{x^{2} \cdot 6(x+2)}[/tex]
[tex]\Rightarrow \frac{30(x+2)}{6 x^{2}(x+2)}$[/tex]
We need to multiply the fraction on the right by [tex]$\frac{x}{x}$[/tex] to obtain a common denominator:
[tex]\frac{x}{x} \times \frac{3}{6 x^{2}+12 x}[/tex]
[tex]\Rightarrow \frac{3 \cdot x}{x\left(6 x^{2}+12 x\right)}[/tex]
[tex]\Rightarrow \frac{3 x}{6 x^{3}+12 x^{2}}[/tex]
[tex]\Rightarrow \frac{3 x}{6 x^{2}(x+2)}$[/tex]
Therefore the least common denominator is [tex]6x^{2} (x+2)[/tex].
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