Let's simplify the given expression step by step:

$c = \sqrt{\left( \sqrt{10} + \sqrt{7} \right)^2 + \left( \sqrt{10} - \sqrt{7} \right)^2}$

Step 1: Expand the squares

First, we'll expand both squared terms:

1. $\left( \sqrt{10} + \sqrt{7} \right)^2 = \left( \sqrt{10} \right)^2 + 2 \times \sqrt{10} \times \sqrt{7} + \left( \sqrt{7} \right)^2 = 10 + 2\sqrt{70} + 7 = 17 + 2\sqrt{70}$
2. $\left( \sqrt{10} - \sqrt{7} \right)^2 = \left( \sqrt{10} \right)^2 - 2 \times \sqrt{10} \times \sqrt{7} + \left( \sqrt{7} \right)^2 = 10 - 2\sqrt{70} + 7 = 17 - 2\sqrt{70}$

Step 2: Add the two terms

Now, add the two expanded expressions:

$\left( 17 + 2\sqrt{70} \right) + \left( 17 - 2\sqrt{70} \right) = 17 + 17 = 34$

Step 3: Take the square root

Finally, take the square root of 34:

$c = \sqrt{34}$

So, the simplified expression for $c$ is:

$c = \sqrt{34}$

Would you like more details or any further steps? Here are 5 related questions you might find useful:

1. What happens if we change the numbers inside the square roots?
2. How can I simplify expressions with nested square roots?
3. What is the geometric interpretation of the expression $\sqrt{(a+b)^2 + (a-b)^2}$?
4. How do you handle square roots of sums and differences?
5. What are some tricks for simplifying expressions involving radicals?

Tip: Always try to simplify square roots and like terms before proceeding with more complex operations.