The problem asks to simplify the expression:

$\frac{\sqrt{27} - \sqrt{24}}{\sqrt{3}}$

Step 1: Simplify the square roots

• $\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}$
• $\sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6}$

So the expression becomes:

$\frac{3\sqrt{3} - 2\sqrt{6}}{\sqrt{3}}$

Step 2: Split the terms

We can rewrite the expression as two separate terms:

$\frac{3\sqrt{3}}{\sqrt{3}} - \frac{2\sqrt{6}}{\sqrt{3}}$

Step 3: Simplify each term

1. For the first term:

$\frac{3\sqrt{3}}{\sqrt{3}} = 3$

2. For the second term, use the property $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$:

$\frac{2\sqrt{6}}{\sqrt{3}} = 2\sqrt{\frac{6}{3}} = 2\sqrt{2}$

Step 4: Final expression

Now the expression simplifies to:

$3 - 2\sqrt{2}$

Conclusion

This matches answer B. $-2\sqrt{2} + 3$, which is the correct choice.

Would you like more details or have any questions?

Here are some related questions:

1. How do you simplify radicals like $\sqrt{27}$ or $\sqrt{24}$?
2. Why does $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$ work?
3. What other strategies are there for simplifying expressions with square roots?
4. Can this process be applied to complex numbers?
5. How does the distributive property relate to simplifying square roots?

Tip: Always look for perfect squares inside the radical to simplify expressions!