https://r2.erweima.ai/mathbot/image/17266191111329huxld87.jpg

The mathematical expression in the image is as follows:

\[
\frac{3}{4 + \frac{1}{3 + \frac{2}{3}}}
\]

Let's break this down step-by-step:

### Step 1: Simplify the innermost fraction
Start by simplifying the fraction inside the denominator:

\[
3 + \frac{2}{3}
\]

To add \( 3 \) and \( \frac{2}{3} \), convert 3 to a fraction:

\[
3 = \frac{9}{3}
\]

Now add:

\[
3 + \frac{2}{3} = \frac{9}{3} + \frac{2}{3} = \frac{11}{3}
\]

### Step 2: Simplify the second part
Now substitute \( 3 + \frac{2}{3} \) with \( \frac{11}{3} \):

\[
4 + \frac{1}{\frac{11}{3}}
\]

The next step is to simplify \( \frac{1}{\frac{11}{3}} \), which is equivalent to multiplying by the reciprocal:

\[
\frac{1}{\frac{11}{3}} = \frac{3}{11}
\]

### Step 3: Simplify the denominator
Now add 4 and \( \frac{3}{11} \):

Convert 4 into a fraction:

\[
4 = \frac{44}{11}
\]

Now add:

\[
4 + \frac{3}{11} = \frac{44}{11} + \frac{3}{11} = \frac{47}{11}
\]

### Step 4: Final expression
Now the expression becomes:

\[
\frac{3}{\frac{47}{11}}
\]

This is the same as multiplying by the reciprocal:

\[
\frac{3}{\frac{47}{11}} = 3 \times \frac{11}{47} = \frac{33}{47}
\]

Thus, the simplified value of the expression is:

\[
\boxed{\frac{33}{47}}
\]

Would you like a detailed explanation on any of these steps?

Here are some related questions for further practice:

1. How do you add fractions with different denominators?
2. What is the general method for simplifying complex fractions?
3. How can you multiply fractions more efficiently?
4. What are some strategies for checking the correctness of fraction operations?
5. Can complex fractions always be simplified into simpler terms?

**Tip:** Always simplify complex fractions step-by-step by working from the innermost expressions outward to avoid mistakes.

Simplify the complex fraction: 3 / (4 + 1 / (3 + 2/3))

This step-by-step guide helps you simplify the complex fraction 3 / (4 + 1 / (3 + 2/3)). Learn how to apply fraction addition, multiplication by reciprocals, and common fraction operations. Suitable for students in Grades 8-10.

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