https://file.aiquickdraw.com/mathbot/file/1730022222853ygmwrzif.jpg

To fully simplify the expression:

\[
\frac{\sqrt{54} + \sqrt{24}}{\sqrt{3}}
\]

we'll follow these steps:

### Step 1: Simplify the square roots in the numerator
1. **Simplify \( \sqrt{54} \):**
   \[
   \sqrt{54} = \sqrt{9 \times 6} = \sqrt{9} \times \sqrt{6} = 3\sqrt{6}
   \]

2. **Simplify \( \sqrt{24} \):**
   \[
   \sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6}
   \]

Now, our expression becomes:
\[
\frac{3\sqrt{6} + 2\sqrt{6}}{\sqrt{3}}
\]

### Step 2: Combine like terms in the numerator
Since both terms have \( \sqrt{6} \) as a common factor:
\[
3\sqrt{6} + 2\sqrt{6} = (3 + 2)\sqrt{6} = 5\sqrt{6}
\]

So the expression is now:
\[
\frac{5\sqrt{6}}{\sqrt{3}}
\]

### Step 3: Simplify the division by \( \sqrt{3} \)
We can break down the square roots:
\[
\frac{5\sqrt{6}}{\sqrt{3}} = 5 \cdot \sqrt{\frac{6}{3}} = 5 \cdot \sqrt{2} = 5\sqrt{2}
\]

### Final Answer:
\[
5\sqrt{2}
\]

Would you like more details on any of these steps?

### Follow-Up Questions
1. How would you simplify an expression with more than two terms inside the square root?
2. What is the process for rationalizing a denominator with a square root?
3. How do we handle square roots with variables inside them?
4. Can this technique be used to simplify cube roots or higher-order roots?
5. What are the properties of radicals that make simplification possible?

#### Tip
When simplifying expressions involving square roots, factor out perfect squares to simplify them more efficiently.

Fully simplify (sqrt(54) + sqrt(24)) / sqrt(3)

Explore a detailed solution to fully simplify (sqrt(54) + sqrt(24)) / sqrt(3) by breaking down the radicals, combining like terms, and rationalizing the denominator. Learn key concepts and properties of square roots. Ideal for Grades 9-10 math students.

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