Let's simplify the given expression:

$\sqrt{24} - 2\sqrt{12} + 7\sqrt{6}$

Step 1: Simplify each square root

• $\sqrt{24}$ can be simplified as: $\sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6}$

• $\sqrt{12}$ can be simplified as: $\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}$

• $\sqrt{6}$ is already in its simplest form.

Step 2: Substitute the simplified terms back into the expression

Now, substitute these back into the original expression:

$2\sqrt{6} - 2(2\sqrt{3}) + 7\sqrt{6}$

Simplify the expression inside the parentheses:

$2\sqrt{6} - 4\sqrt{3} + 7\sqrt{6}$

Step 3: Combine like terms

Combine the terms involving $\sqrt{6}$:

$(2\sqrt{6} + 7\sqrt{6}) - 4\sqrt{3} = 9\sqrt{6} - 4\sqrt{3}$

Final Answer

The simplified form of the expression is:

$9\sqrt{6} - 4\sqrt{3}$

Would you like further details on any part of this process?

Here are some related questions you might find useful:

1. How do you simplify square roots in algebraic expressions?
2. What are the rules for combining like terms involving square roots?
3. How can we factor square roots to simplify them further?
4. What happens when there are different square roots in a single expression?
5. How do you multiply and divide terms involving square roots?
6. Can square roots be negative, and how does that affect simplification?
7. How does this simplification change if coefficients are added or subtracted from the square root terms?
8. Can you provide examples of simplifying more complex square root expressions?

Tip: Always check if the square roots can be simplified by factoring them into perfect squares before combining like terms.