The mathematical expression shown in the image is quite complex, involving several nested fractions. Here's how it is structured:

$1 + \frac{1}{1 + \frac{1}{3 + \frac{1}{3 + \frac{1}{3}}}}$

Step-by-Step Simplification:

1. Start with the innermost fraction: $\frac{1}{3}$ This remains as it is since $\frac{1}{3} = 0.3333$.

2. The next part is: $3 + \frac{1}{3} = 3 + 0.3333 = 3.3333$

3. Now, substitute that into the next fraction: $\frac{1}{3 + \frac{1}{3}} = \frac{1}{3.3333} = 0.3$

4. Moving to the next level: $3 + 0.3 = 3.3$

5. Then substitute back into the next fraction: $\frac{1}{3.3} \approx 0.303$

6. Finally, handle the outermost part: $1 + 0.303 = 1.303$

So, the final result is approximately $1.303$.

Would you like further details or have any questions?

Related Questions:

1. How are continued fractions simplified systematically?
2. What are the practical applications of continued fractions?
3. Can we express irrational numbers using continued fractions?
4. How would this expression change if we altered one of the numbers?
5. Why are nested fractions useful in number theory?

Tip: When working with nested fractions, always begin simplification from the innermost term and proceed outward.