Question:
Which shows the rational expression written using the least common denominator?
[tex]$\frac{x-2}{x^3} +\frac{x+1}{2x^3}[/tex]
Answer:
[tex]$\frac{3(x-1)}{2x^3}[/tex]
Solution:
Given expression is
[tex]$\frac{x-2}{x^3} +\frac{x+1}{2x^3}[/tex]
Denominators of the fraction are [tex]x^3[/tex] and [tex]2x^3[/tex].
Factors of [tex]x^3=x\times x\times x[/tex]
Factors of [tex]2x^3=2\times x \times x \times x[/tex]
Least common factors = [tex]x\times x\times x \times2=2x^3[/tex]
Now make the denominators same using LCM.
[tex]$\frac{x-2}{x^3} +\frac{x+1}{2x^3}=\frac{(x-2)\times 2}{x^3 \times 2} +\frac{x+1}{2x^3}[/tex]
[tex]$=\frac{2x-4}{2x^3} +\frac{x+1}{2x^3}[/tex]
[tex]$=\frac{2x-4+x+1}{2x^3}[/tex]
[tex]$=\frac{3x-3}{2x^3}[/tex]
[tex]$=\frac{3(x-1)}{2x^3}[/tex]
Hence the answer is [tex]$\frac{3(x-1)}{2x^3}[/tex].
