https://file.aiquickdraw.com/mathbot/file/172565141454818e4j2w3.jpg

The expression shown in the image is:

\[
\frac{2a + 3b}{\frac{1}{a} + \frac{2}{b}}
\]

Let's simplify this step by step.

### Step 1: Simplify the denominator
The denominator is the sum of two fractions: \(\frac{1}{a} + \frac{2}{b}\). To add them, we need a common denominator.

\[
\frac{1}{a} + \frac{2}{b} = \frac{b}{ab} + \frac{2a}{ab} = \frac{b + 2a}{ab}
\]

So, the expression now becomes:

\[
\frac{2a + 3b}{\frac{b + 2a}{ab}} 
\]

### Step 2: Simplify the entire expression
When dividing by a fraction, we multiply by its reciprocal. So we have:

\[
\frac{2a + 3b}{\frac{b + 2a}{ab}} = (2a + 3b) \times \frac{ab}{b + 2a}
\]

Now, notice that \(b + 2a\) in the denominator is the same as \(2a + b\) in the numerator (just the terms in a different order). Therefore, the two terms cancel out:

\[
(2a + 3b) \times \frac{ab}{b + 2a} = 3b \cdot ab = 3ab
\]

Thus, the simplified expression is:

\[
3ab
\]

Let me know if you need further details or have any questions!

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Here are 5 related questions that you may find helpful:

1. How do you add fractions with different denominators?
2. What is the difference between simplifying an expression and solving an equation?
3. How do you multiply and divide algebraic expressions?
4. What is the process for finding the reciprocal of a fraction?
5. Can this technique be applied to more complex rational expressions?

**Tip**: Always look for common factors in both the numerator and the denominator when simplifying algebraic fractions. This helps to reduce the expression faster!

Simplify the expression (2a + 3b) / (1/a + 2/b).

Learn how to simplify the algebraic expression (2a + 3b) / (1/a + 2/b) using fraction addition, reciprocals, and multiplication. This problem covers key algebraic concepts like rational expressions and fraction operations, suitable for students in Grades 9-10.

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