Let's break down the problem and simplify the expression step-by-step. The given expression is:

$4\sqrt{3} + 3\sqrt{12} - \sqrt{27}.$

Step 1: Simplify Each Square Root

1. $\sqrt{12}$: $\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}.$

2. $\sqrt{27}$: $\sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3\sqrt{3}.$

Step 2: Substitute the Simplified Square Roots into the Expression

$4\sqrt{3} + 3\sqrt{12} - \sqrt{27} = 4\sqrt{3} + 3 \times 2\sqrt{3} - 3\sqrt{3}.$

Step 3: Simplify the Expression

$4\sqrt{3} + 6\sqrt{3} - 3\sqrt{3}.$

Combine the terms: $(4 + 6 - 3) \sqrt{3} = 7\sqrt{3}.$

So, the simplified form of the expression is $7\sqrt{3}$.

Answer

The correct option is: $\text{b. } 7\sqrt{3}.$

Would you like a detailed explanation of any step or have further questions?

Here are 5 related questions for further practice:

1. Simplify $5\sqrt{8} + 2\sqrt{18}$.
2. What is the simplified form of $3\sqrt{50} - 2\sqrt{8} + \sqrt{32}$?
3. If $x = \sqrt{75} + \sqrt{48}$, what is $x$ in its simplest form?
4. How do you simplify $6\sqrt{27} + 4\sqrt{12} - 5\sqrt{3}$?
5. Solve for $x$ if $2x = 4\sqrt{5} - 2\sqrt{20}$.

Tip: Always look for square factors inside a square root to simplify it! For example, $\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$.