$$ 5\sqrt2 $$$$  $$$$ +2\sqrt{\frac72} $$

To simplify the expression $ 5\sqrt{2} + 2\sqrt{\frac{7}{2}} $, let's break down each term and simplify if possible.

1. **Simplify each term:**

   - The term $ 5\sqrt{2} $ is already in its simplest form.
   - For the term $ 2\sqrt{\frac{7}{2}} $, we can simplify it further:
   
   \[
   2\sqrt{\frac{7}{2}} = 2 \times \frac{\sqrt{7}}{\sqrt{2}}
   \]

2. **Simplify the fraction under the square root:**

   \[
   2 \times \frac{\sqrt{7}}{\sqrt{2}} = 2 \times \frac{\sqrt{7}}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = 2 \times \frac{\sqrt{14}}{2} = \sqrt{14}
   \]

   Now, the expression simplifies to:

   \[
   5\sqrt{2} + \sqrt{14}
   \]

3. **Conclusion:**

   The expression $ 5\sqrt{2} + 2\sqrt{\frac{7}{2}} $ simplifies to:

   \[
   \boxed{5\sqrt{2} + \sqrt{14}}
   \]

This is the simplest form of the given expression.

Would you like more details or have any questions?

Here are 5 related questions to further explore the topic:

1. How do you simplify expressions involving square roots of fractions?
2. What are some common rules for simplifying square root expressions?
3. Can you explain the concept of like and unlike terms in expressions involving square roots?
4. How can you rationalize a denominator involving a square root?
5. What are some practical applications of simplifying square root expressions in real-life problems?

**Tip:** When simplifying square root expressions, look for opportunities to factor out squares or simplify fractions under the square root to make the expression easier to handle.

Learn how to simplify the expression 5√2 + 2√(7/2) step-by-step with detailed explanations. This problem involves simplifying square roots and fractional exponents, suitable for high school students.

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