Let's simplify the expression $7\sqrt{2} + \sqrt{8} + \sqrt{18}$.

Step 1: Simplify $\sqrt{8}$

We know that $\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}$.

Step 2: Simplify $\sqrt{18}$

Similarly, $\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}$.

Step 3: Substitute the simplified values into the expression

Now, substitute the simplified values of $\sqrt{8}$ and $\sqrt{18}$ into the original expression:

$7\sqrt{2} + 2\sqrt{2} + 3\sqrt{2}$

Step 4: Combine like terms

Since all terms contain $\sqrt{2}$, we can combine the coefficients:

$(7 + 2 + 3)\sqrt{2} = 12\sqrt{2}$

Final Answer:

$7\sqrt{2} + \sqrt{8} + \sqrt{18} = 12\sqrt{2}$

Would you like further details or have any questions?

Related Questions:

1. How do you simplify other square roots like $\sqrt{12}$ or $\sqrt{50}$?
2. What is the general method for simplifying square roots with products?
3. How would you handle expressions with cube roots instead of square roots?
4. What are the properties of square roots that allow us to combine like terms?
5. Can you simplify $\sqrt{a} + \sqrt{b}$ if $a$ and $b$ are not perfect squares?

Tip:

When simplifying square roots, always factor the number inside the root to see if it contains any perfect squares!