To simplify the given expression [tex]\(\frac{4 x}{3 a b^3}+\frac{2 y}{4 a^2 b^2}\)[/tex], we will follow these steps:
1. Simplify each term individually, if possible.
2. Find a common denominator for the fractions.
3. Combine the fractions into a single expression.
4. Simplify the resulting expression.
Let's start by simplifying each term.
The first term is [tex]\(\frac{4 x}{3 a b^3}\)[/tex].
The second term is [tex]\(\frac{2 y}{4 a^2 b^2}\)[/tex]. Notice that [tex]\(\frac{2 y}{4}\)[/tex] simplifies to [tex]\(\frac{y}{2}\)[/tex]. So, we can rewrite the second term as [tex]\(\frac{y}{2 a^2 b^2}\)[/tex].
Now the expression is:
[tex]\[
\frac{4 x}{3 a b^3} + \frac{y}{2 a^2 b^2}
\][/tex]
Next, we need to find a common denominator for the two fractions. The denominators are [tex]\(3 a b^3\)[/tex] and [tex]\(2 a^2 b^2\)[/tex]. The least common multiple (LCM) of these denominators is:
[tex]\[
\text{LCM}(3 a b^3, 2 a^2 b^2) = 6 a^2 b^3
\][/tex]
Now we convert each fraction to have this common denominator.
For the first term [tex]\(\frac{4 x}{3 a b^3}\)[/tex], we need to multiply the numerator and the denominator by [tex]\(2 a\)[/tex]:
[tex]\[
\frac{4 x \cdot 2 a}{3 a b^3 \cdot 2 a} = \frac{8 a x}{6 a^2 b^3}
\][/tex]
For the second term [tex]\(\frac{y}{2 a^2 b^2}\)[/tex], we need to multiply the numerator and the denominator by [tex]\(3 b\)[/tex]:
[tex]\[
\frac{y \cdot 3 b}{2 a^2 b^2 \cdot 3 b} = \frac{3 b y}{6 a^2 b^3}
\][/tex]
Now the expression is:
[tex]\[
\frac{8 a x}{6 a^2 b^3} + \frac{3 b y}{6 a^2 b^3}
\][/tex]
Since both fractions now have the same denominator, we can combine them:
[tex]\[
\frac{8 a x + 3 b y}{6 a^2 b^3}
\][/tex]
This is the simplified form of the given expression. Therefore, the final answer is:
[tex]\[
\frac{8 a x + 3 b y}{6 a^2 b^3}
\][/tex]
