91Ó°ÊÓ

When working with algebraic expressions, one important skill is the ability to combine like terms. These are terms that have the same variable part. In the exercise we see the terms \(-5\sqrt{3}\), \(2\sqrt{3}\), and \(-3\sqrt{3}\). Here, the "like" part is the \(\sqrt{3}\).
To combine them, you simply add or subtract their coefficients, which are the numbers in front of the square root.
- Start with \(-5 + 2 - 3\). - Adding and subtracting these gives \(-6\).So, putting it all together, you have \(-6\sqrt{3}\). By mastering this technique, you streamline expressions and make calculations much easier.

Prime factorization is a useful method for breaking down numbers into their basic building blocks: prime numbers. This is essential when simplifying square roots.
Consider the number 75. We can express it as the product of 25 and 3. Next, break 25 down to its prime factors, which is 5 multiplied by 5. This makes "75" expressible as \(5^2 \times 3\). This neatly translates to \(\sqrt{75} = \sqrt{5^2 \times 3}\). Other examples include number 12, which can be broken down to 4 and 3, or the products \(2^2 \times 3\), simplifying to \(\sqrt{12} = \sqrt{2^2 \times 3}\).
By rewriting numbers this way, finding square roots becomes straightforward and manageable, helping to further simplify the given expressions.

Algebraic expressions are combinations of numbers, variables, and operations. In the given exercise, the expression \(-\sqrt{75} + \sqrt{12} - 3\sqrt{3}\) serves as our algebraic expression.
Each term appears to have a numerical part (coefficient) paired with a variable part. The variable parts aren't always just 'x' or 'y'—sometimes, they can be something like \(\sqrt{3}\), as seen in this exercise.
When working with such expressions, it's crucial to simplify them as much as possible. This often means carefully addressing each part of the expression individually, as seen by simplifying the roots and, following that, combining terms that are alike. A simplified algebraic expression is not only neater, but also easier to interpret and use in further calculations.