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Part 1
Simplify.
3 StartRoot 48 EndRoot plus StartRoot 12 EndRoot minus StartRoot 32 EndRoot

Let's simplify the expression:

\[
3 \sqrt{48} + \sqrt{12} - \sqrt{32}
\]

### Step 1: Simplify each square root term.

1. **\(\sqrt{48}\):**

   Break \(48\) down into its prime factors:
   \[
   48 = 16 \times 3
   \]
   Now, \(\sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3}\).

2. **\(\sqrt{12}\):**

   Break \(12\) down into its prime factors:
   \[
   12 = 4 \times 3
   \]
   Now, \(\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}\).

3. **\(\sqrt{32}\):**

   Break \(32\) down into its prime factors:
   \[
   32 = 16 \times 2
   \]
   Now, \(\sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2}\).

### Step 2: Substitute the simplified square roots into the original expression.

\[
3 \sqrt{48} + \sqrt{12} - \sqrt{32} = 3(4\sqrt{3}) + 2\sqrt{3} - 4\sqrt{2}
\]

Now, simplify the terms with \(\sqrt{3}\):

\[
12\sqrt{3} + 2\sqrt{3} - 4\sqrt{2}
\]

\[
(12\sqrt{3} + 2\sqrt{3}) = 14\sqrt{3}
\]

So, the expression becomes:

\[
14\sqrt{3} - 4\sqrt{2}
\]

### Final Simplified Expression:

\[
14\sqrt{3} - 4\sqrt{2}
\]

Would you like a more detailed explanation or have any questions about this solution?

Here are some related questions for further exploration:
1. How do you simplify square roots with prime factorization?
2. Can you explain how to combine like terms with square roots?
3. What are some tips for simplifying expressions involving square roots?
4. How can you simplify other expressions with multiple square roots?
5. How would you approximate the values of \(\sqrt{3}\) and \(\sqrt{2}\)?

**Tip:** When simplifying square roots, always look for perfect squares in the radicand (the number inside the square root) to simplify them efficiently.

Simplify. 3 StartRoot 48 EndRoot plus StartRoot 12 EndRoot minus StartRoot 32 EndRoot

Learn how to simplify the expression 3√48 + √12 - √32 with a detailed explanation. This problem involves simplifying square roots using prime factorization and combining like terms. Suitable for students in Grades 6-8.

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