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Square roots can sometimes seem tricky, but breaking them down makes them easier to grasp. A square root is a number that, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3, because 3 multiplied by 3 is 9. When faced with more complex numbers like 12, we can simplify square roots by identifying their factor components that are perfect squares. Let's consider \(\sqrt{12}\). Breaking down 12 into its prime factors gives \(2^2 * 3\). By defining these factors, we notice that \(2^2\) is a perfect square, so the square root of \(2^2\) is 2. Therefore, \(\sqrt{12}\) turns into \(2\sqrt{3}\). Simplifying square roots like this can turn a complex expression into something more manageable.

The distributive property is a fundamental algebraic concept that allows us to multiply a single term across terms within parentheses. To put it simply, if you have \(a(b + c)\), the distributive property lets you transform it into \(ab + ac\). This step-by-step procession is essential to simplifying equations correctly.In our exercise, we have the expression \(3(2\sqrt{3} - \sqrt{3})\). Here, we apply the distributive property by multiplying 3 with each term inside the parenthesis:

• First, multiply \(3\) and \(2\sqrt{3}\) to get \(6\sqrt{3}\).
• Then, multiply \(3\) by \(\sqrt{3}\) to obtain \(3\sqrt{3}\).

This breaks down the expression into easier parts to handle before moving forward.

Simplifying algebraic expressions is about making them easier to read and use. One common task in algebraic simplification is combining like terms, which are terms with the same variable raised to the same power. This is a straightforward process once you recognize these terms.Looking at the expression \(6\sqrt{3} - 3\sqrt{3}\), both terms are like terms, as they both contain \(\sqrt{3}\). We can simplify by subtracting the coefficients \(6\) and \(3\), resulting in \(3\sqrt{3}\). This is the simplified form of the expression, efficiently condensing it to its simplest version, thus making further calculations or interpretations clearer and easier.