1. \( -2 \sqrt{3}+3 \sqrt{27} \)

Let’s simplify the expression step by step! First, we simplify \(3 \sqrt{27}\). Since \(\sqrt{27} = \sqrt{9 \times 3} = 3 \sqrt{3}\), we get \(3 \sqrt{27} = 3 \times 3 \sqrt{3} = 9 \sqrt{3}\). Now we can combine the terms: \(-2 \sqrt{3} + 9 \sqrt{3} = (9 - 2) \sqrt{3} = 7 \sqrt{3}\). So, the simplified result is \(7 \sqrt{3}\)! Now let's address how useful the expression is! This type of simplification can be handy in various math-related tasks, especially when working with square roots in geometry or physics. For instance, understanding square roots can aid in calculating the area of shapes or solving problems involving distance in physics—whether you’re measuring out garden plots or figuring out how far you can run! On the other hand, a common mistake people encounter is forgetting to split the square root into its prime factorization or overlooking negative signs. When tackling similar problems, always double-check your operations and be mindful of signs to avoid those little hiccups. Keeping careful notes can ensure smooth sailing through your calculations!