We are asked to multiply the following expression:

$\sqrt{7} \cdot 7(9 - 10\sqrt{7})$

Step 1: Distribute $\sqrt{7} \cdot 7$ across the parentheses

Distribute $\sqrt{7} \cdot 7$ to each term inside the parentheses:

$\sqrt{7} \cdot 7 \cdot 9 - \sqrt{7} \cdot 7 \cdot 10\sqrt{7}$

Step 2: Simplify each term

• The first term:
  $\sqrt{7} \cdot 7 \cdot 9 = 7 \cdot 9 \cdot \sqrt{7} = 63\sqrt{7}$

• The second term:
  $\sqrt{7} \cdot 7 \cdot 10\sqrt{7} = 7 \cdot 10 \cdot \sqrt{7} \cdot \sqrt{7} = 70 \cdot 7 = 490$

Step 3: Combine the results

Now, we combine both terms:

$63\sqrt{7} - 490$

Thus, the result is:

$63\sqrt{7} - 490$

Would you like further explanation or help with any additional steps?

Related Questions:

1. What happens if we multiply expressions with square roots like $\sqrt{a} \cdot \sqrt{b}$?
2. How can we simplify the expression $\sqrt{a} \cdot \sqrt{a}$?
3. What are some strategies to simplify more complicated square root expressions in algebra?
4. How does distributing in multiplication affect the outcome in expressions involving parentheses?
5. How can you recognize when an expression can be factored further?

Tip: When multiplying square roots, remember $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$.