Let's simplify the given expression step by step:

$3\sqrt{12} + \frac{\sqrt{16}}{3} - \sqrt{27} + \frac{\sqrt{27}}{3}$

Step 1: Simplify each term

• First term: $3\sqrt{12}$

We can break $\sqrt{12}$ into prime factors: $\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$ Now multiply by 3: $3\sqrt{12} = 3 \times 2\sqrt{3} = 6\sqrt{3}$

• Second term: $\frac{\sqrt{16}}{3}$

Since $\sqrt{16} = 4$, we have: $\frac{\sqrt{16}}{3} = \frac{4}{3}$

• Third term: $\sqrt{27}$

We can break $\sqrt{27}$ into prime factors: $\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}$

• Fourth term: $\frac{\sqrt{27}}{3}$

We already know that $\sqrt{27} = 3\sqrt{3}$, so: $\frac{\sqrt{27}}{3} = \frac{3\sqrt{3}}{3} = \sqrt{3}$

Step 2: Combine all terms

Now, we combine all the simplified terms: $6\sqrt{3} + \frac{4}{3} - 3\sqrt{3} + \sqrt{3}$

Combine the $\sqrt{3}$ terms: $6\sqrt{3} - 3\sqrt{3} + \sqrt{3} = (6 - 3 + 1)\sqrt{3} = 4\sqrt{3}$

Thus, the expression becomes: $4\sqrt{3} + \frac{4}{3}$

Final Answer:

$4\sqrt{3} + \frac{4}{3}$

This is the simplified exact answer.

Would you like more details on any step or have other questions? Here are a few related questions for further practice:

1. How do you simplify expressions with square roots involving prime factorization?
2. What is the distributive property, and how does it apply to expressions with square roots?
3. Can you simplify $\sqrt{50} + \frac{\sqrt{9}}{2}$?
4. How do you handle fractional square roots, like $\frac{\sqrt{16}}{5}$?
5. What is the difference between simplifying an expression and evaluating it numerically?

Tip: Always factor numbers under square roots to simplify them before performing further calculations!