3\sqrt(54)-2\sqrt(24)-\sqrt(96)+4\sqrt(63)

Let's simplify the given expression step by step:

\[
3\sqrt{54} - 2\sqrt{24} - \sqrt{96} + 4\sqrt{63}
\]

### Step 1: Simplify each square root term.
We begin by factoring the numbers inside the square roots.

- \( \sqrt{54} = \sqrt{9 \times 6} = 3\sqrt{6} \)
- \( \sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6} \)
- \( \sqrt{96} = \sqrt{16 \times 6} = 4\sqrt{6} \)
- \( \sqrt{63} = \sqrt{9 \times 7} = 3\sqrt{7} \)

Now, substitute these simplified square roots into the expression:

\[
3(3\sqrt{6}) - 2(2\sqrt{6}) - 4\sqrt{6} + 4(3\sqrt{7})
\]

### Step 2: Multiply the coefficients.
Perform the multiplications:

\[
9\sqrt{6} - 4\sqrt{6} - 4\sqrt{6} + 12\sqrt{7}
\]

### Step 3: Combine like terms.
Combine the terms with \( \sqrt{6} \):

\[
(9 - 4 - 4)\sqrt{6} + 12\sqrt{7}
\]

This simplifies to:

\[
1\sqrt{6} + 12\sqrt{7}
\]

### Final Answer:
The simplified form of the expression is:

\[
\sqrt{6} + 12\sqrt{7}
\]

Would you like more details or have any questions?

Here are 5 related questions to expand on this:

1. How do you simplify radicals?
2. What are some examples of irrational numbers like \( \sqrt{6} \)?
3. How do coefficients in front of square roots affect the simplification process?
4. Can you explain the properties of square roots for multiplying and factoring?
5. What happens when you combine terms with different square roots?

**Tip:** When simplifying square roots, always look for perfect square factors to make the process easier!

This guide walks you through simplifying the expression 3√(54) - 2√(24) - √(96) + 4√(63) by factoring square roots and combining like terms. Learn about simplifying radicals and properties of square roots, suitable for high school students in Grades 9-12.

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