To solve the given problem, we need to simplify the expression:

$\frac{\sqrt{15} + \sqrt{35} + \sqrt{21} + 5}{\sqrt{3} + 2\sqrt{5} + \sqrt{7}}$

Let's first look at simplifying the numerator and the denominator separately.

Step 1: Simplify the Numerator

The numerator is:

$\sqrt{15} + \sqrt{35} + \sqrt{21} + 5$

There is no straightforward way to combine the square roots, so we consider their values directly:

• $\sqrt{15}$ remains $\sqrt{15}$
• $\sqrt{35}$ remains $\sqrt{35}$
• $\sqrt{21}$ remains $\sqrt{21}$
• $5$ remains $5$

The numerator simplifies to:

$\sqrt{15} + \sqrt{35} + \sqrt{21} + 5$

Step 2: Simplify the Denominator

The denominator is:

$\sqrt{3} + 2\sqrt{5} + \sqrt{7}$

Again, there is no straightforward way to combine the square roots, so we consider their values directly:

• $\sqrt{3}$ remains $\sqrt{3}$
• $2\sqrt{5}$ remains $2\sqrt{5}$
• $\sqrt{7}$ remains $\sqrt{7}$

The denominator simplifies to:

$\sqrt{3} + 2\sqrt{5} + \sqrt{7}$

Step 3: Combine Numerator and Denominator

Now we combine the simplified numerator and denominator:

$\frac{\sqrt{15} + \sqrt{35} + \sqrt{21} + 5}{\sqrt{3} + 2\sqrt{5} + \sqrt{7}}$

Step 4: Check for Any Further Simplification

It is not immediately obvious if there is a simple rational number or a known algebraic simplification that results from this division. However, we can test whether the numerator and the denominator share a common factor by calculating their approximate numerical values:

Using approximate values:

• $\sqrt{3} \approx 1.732$
• $2\sqrt{5} \approx 4.472$
• $\sqrt{7} \approx 2.646$
• $\sqrt{15} \approx 3.873$
• $\sqrt{35} \approx 5.916$
• $\sqrt{21} \approx 4.583$

Approximate values: $\sqrt{3} + 2\sqrt{5} + \sqrt{7} \approx 1.732 + 4.472 + 2.646 \approx 8.85$ $\sqrt{15} + \sqrt{35} + \sqrt{21} + 5 \approx 3.873 + 5.916 + 4.583 + 5 \approx 19.372$

Using these approximations: $\frac{19.372}{8.85} \approx 2.19$

Conclusion

The expression simplifies to approximately 2. The precise value would require exact simplification techniques, possibly hinting at some algebraic manipulation or insight into the structure of the problem.

Would you like more details or have any questions about this problem?

Here are five questions you can ask:

1. How do I simplify complex fractions involving square roots?
2. Can you solve another example of a similar fraction?
3. What are some techniques for approximating irrational numbers?
4. How do I rationalize the denominator in expressions involving square roots?
5. Can you explain how to solve olympiad-level math problems in general?

Tip: When dealing with complex fractions involving square roots, look for patterns or potential factorization to simplify the expression more easily.