Let's break down the expression step by step:

We are tasked with simplifying:

$4\sqrt{7} \left( \sqrt{2} - 5\sqrt{10} \right)$

Step 1: Distribute the $4\sqrt{7}$

We will distribute $4\sqrt{7}$ to each term inside the parentheses.

• Multiply $4\sqrt{7}$ by $\sqrt{2}$
• Multiply $4\sqrt{7}$ by $-5\sqrt{10}$

Step 2: Multiply the first term

First, simplify $4\sqrt{7} \times \sqrt{2}$:

$4\sqrt{7} \times \sqrt{2} = 4 \times \sqrt{7 \times 2} = 4\sqrt{14}$

Step 3: Multiply the second term

Next, simplify $4\sqrt{7} \times (-5\sqrt{10})$:

$4\sqrt{7} \times (-5\sqrt{10}) = -20 \times \sqrt{7 \times 10} = -20\sqrt{70}$

Step 4: Combine the results

Now, combining both terms, we have:

$4\sqrt{14} - 20\sqrt{70}$

This is the simplified form of the given expression.

Final Answer:

$4\sqrt{14} - 20\sqrt{70}$

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Let me know if you want further clarification or a deeper explanation on any step.

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Further Questions for Practice:

1. How would you simplify $\sqrt{48}$?
2. What happens when you multiply two square roots, like $\sqrt{3} \times \sqrt{5}$?
3. How do you rationalize a denominator involving a square root?
4. What is the difference between simplifying a radical and expanding an expression with square roots?
5. Can you simplify $3\sqrt{8} + 2\sqrt{18}$?

Tip: When multiplying square roots, always combine the numbers inside the roots first before simplifying!