Question

unit: rational expressions
progress:
the movement of the progress bar may be uneven because questions can be worth more or less (incl
add. state the sum in simplest form.
\\(\frac{2a - 3}{10a^3b^2} + \frac{3}{15a^2b^2}\\)
\\(\circ\\) \\(\frac{4a - 3}{10a^3b^2}\\)
\\(\circ\\) \\(\frac{2a}{150a^5b^4}\\)
\\(\circ\\) \\(\frac{12a - 9}{30a^3b^2}\\)
\\(\circ\\) \\(\frac{2a}{10a^3b^2 + 15a^2b^2}\\)

Explanation:

Step1: Find the LCD of denominators

The denominators are \(10a^{3}b^{2}\) and \(15a^{2}b^{2}\). Prime factorize: \(10 = 2\times5\), \(15 = 3\times5\); for \(a\): highest power is \(a^{3}\); for \(b\): highest power is \(b^{2}\). So LCD is \(2\times3\times5\times a^{3}b^{2}=30a^{3}b^{2}\).

Step2: Rewrite fractions with LCD

Rewrite \(\frac{2a - 3}{10a^{3}b^{2}}\) as \(\frac{(2a - 3)\times3}{30a^{3}b^{2}}=\frac{6a - 9}{30a^{3}b^{2}}\).
Rewrite \(\frac{3}{15a^{2}b^{2}}\) as \(\frac{3\times2a}{30a^{3}b^{2}}=\frac{6a}{30a^{3}b^{2}}\).

Step3: Add the fractions

Add the numerators: \(\frac{6a - 9 + 6a}{30a^{3}b^{2}}=\frac{12a - 9}{30a^{3}b^{2}}\) (we can also factor numerator: \(12a - 9 = 3(4a - 3)\), but \(\frac{12a - 9}{30a^{3}b^{2}}\) is equivalent and one of the options).

Answer:

\(\frac{12a - 9}{30a^{3}b^{2}}\) (corresponding to the option \(\boldsymbol{\frac{12a - 9}{30a^{3}b^{2}}}\))