The problem shown is to simplify the expression:

$\sqrt{5} \left( 3\sqrt{10} + \sqrt{7} \right)$

Let’s solve step by step:

Step 1: Distribute $\sqrt{5}$ to both terms inside the parentheses:

$\sqrt{5} \cdot 3\sqrt{10} + \sqrt{5} \cdot \sqrt{7}$

Step 2: Simplify each term:

1. $\sqrt{5} \cdot 3\sqrt{10}$: Combine the square roots: $3 \cdot \sqrt{5 \cdot 10} = 3 \cdot \sqrt{50}$ Simplify $\sqrt{50}$ as $\sqrt{25 \cdot 2} = 5\sqrt{2}$. Thus: $3 \cdot \sqrt{50} = 15\sqrt{2}$

2. $\sqrt{5} \cdot \sqrt{7}$: Combine the square roots: $\sqrt{5 \cdot 7} = \sqrt{35}$

Step 3: Combine the simplified terms:

$15\sqrt{2} + \sqrt{35}$

This is the simplified expression.

Let me know if you'd like further clarification! 😊

Here are 5 related questions for practice:

1. Simplify $\sqrt{3} \cdot \sqrt{12}$.
2. Expand $(\sqrt{2} + \sqrt{5})^2$.
3. Simplify $\sqrt{8} + \sqrt{18}$.
4. Distribute $2\sqrt{3}$ across $(\sqrt{6} + \sqrt{2})$.
5. Simplify $\sqrt{20} + \sqrt{45}$.

Tip: Remember that combining radicals requires the same "radicand" (number inside the square root), much like combining like terms in algebra!