\frac{5a+2}{a^2}+\frac{3}{a}-\frac{4}{3a}

We are given the expression:

\[
\frac{5a+2}{a^2} + \frac{3}{a} - \frac{4}{3a}
\]

To simplify, we need to express all terms with the same denominator. The least common denominator (LCD) of these fractions is \( 3a^2 \). Let's rewrite each term with this denominator:

1. For \( \frac{5a+2}{a^2} \), multiply both the numerator and denominator by 3 to get:

\[
\frac{3(5a+2)}{3a^2} = \frac{15a+6}{3a^2}
\]

2. For \( \frac{3}{a} \), multiply both the numerator and denominator by 3a to get:

\[
\frac{3 \cdot 3a}{3a^2} = \frac{9a}{3a^2}
\]

3. For \( -\frac{4}{3a} \), multiply both the numerator and denominator by \( a \) to get:

\[
-\frac{4a}{3a^2}
\]

Now we can combine all three fractions over the common denominator \( 3a^2 \):

\[
\frac{15a+6}{3a^2} + \frac{9a}{3a^2} - \frac{4a}{3a^2} = \frac{15a + 6 + 9a - 4a}{3a^2}
\]

Simplify the numerator:

\[
\frac{(15a + 9a - 4a) + 6}{3a^2} = \frac{20a + 6}{3a^2}
\]

So the simplified form of the expression is:

\[
\frac{20a + 6}{3a^2}
\]

If you want, we can factor the numerator as well:

\[
\frac{2(10a + 3)}{3a^2}
\]

Would you like further details on any step or have any other questions?

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Here are 5 related questions:

1. How do you find the least common denominator in algebraic fractions?
2. What are the steps to add algebraic fractions?
3. How do you simplify expressions involving fractions and variables?
4. How can you factor polynomials with common factors?
5. How do negative exponents affect the simplification of algebraic fractions?

**Tip:** Always look for common denominators when adding or subtracting fractions. It makes the process of combining them much easier.

\frac{5a+2}{a^2} + \frac{3}{a} - \frac{4}{3a}

Explore how to simplify algebraic fractions involving variables with a step-by-step solution. This problem covers key concepts like finding the least common denominator (LCD), combining fractions, and factoring. Suitable for Grades 8-10 students.

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