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Simplify the expression:

[tex]\[ \frac{4x}{3ab^3} + \frac{2y}{4a^2b^2} \][/tex]

Sagot :

GinnyAnswer
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]
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