Question

table of contents > module 7: rational expressions and equations > practice: add and subtract
practice: add and subtract rationa
part i
practice: add and subtract rational expressions par
score: 5/6 answered: 5/6
question 6
add \\(\frac{-4a}{14m^2} + \frac{-9a}{7m}\\) and state the difference in simplest form.
the domain is \\(m \
eq \square\\)

Explanation:

Step1: Find the common denominator

The denominators are \(14m^2\) and \(7m\). The least common denominator (LCD) of \(14m^2\) and \(7m\) is \(14m^2\) (since \(14m^2 = 2\times7\times m\times m\) and \(7m=7\times m\), so the LCD takes the highest power of each factor: \(2\times7\times m^2 = 14m^2\)).

For the second fraction \(\frac{-9a}{7m}\), we multiply the numerator and denominator by \(2m\) to get the denominator \(14m^2\):
\(\frac{-9a}{7m}=\frac{-9a\times2m}{7m\times2m}=\frac{-18am}{14m^2}\)

Step2: Add the fractions

Now we have \(\frac{-4a}{14m^2}+\frac{-18am}{14m^2}\). Since the denominators are the same, we add the numerators:
\(\frac{-4a - 18am}{14m^2}\)
We can factor out \(-2a\) from the numerator:
\(\frac{-2a(2 + 9m)}{14m^2}\)
Simplify the fraction by dividing numerator and denominator by \(2\):
\(\frac{-a(2 + 9m)}{7m^2}=\frac{-2a - 9am}{7m^2}\) (or we can also write it as \(\frac{-a(9m + 2)}{7m^2}\))

Step3: Find the domain

The original fractions are undefined when the denominator is zero. For \(\frac{-4a}{14m^2}\), denominator \(14m^2 = 0\) when \(m = 0\). For \(\frac{-9a}{7m}\), denominator \(7m=0\) when \(m = 0\). So the domain is \(m
eq0\).

Answer:

The sum in simplest form is \(\boldsymbol{\frac{-2a - 9am}{7m^2}}\) (or \(\boldsymbol{\frac{-a(9m + 2)}{7m^2}}\)) and the domain is \(m
eq\boldsymbol{0}\).