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Square roots are a fundamental concept in algebra. When we talk about the square root of a number, we're looking for a value that, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3 because 3 multiplied by itself is 9. In expressions like \(4 \sqrt{45} - 9 \sqrt{5}\), it's often possible to simplify square roots further. By identifying factors within the number under the square root, particularly perfect squares like 4, 9, 16, and so on, we can simplify. In the example, \(\sqrt{45}\) was rewritten as \(\sqrt{9 \times 5}\). Since \(\sqrt{9}\) is 3, this allowed us to simplify \(\sqrt{45}\) into \(3\sqrt{5}\). Simplifying square roots helps in making the entire expression cleaner and often required when dealing with algebraic expressions.

Combining like terms is a technique used in algebra to simplify expressions. Like terms refer to terms that have the same variables raised to the same power. In the expression involving square roots, like terms have the same radical part. For example, in our expression \(12\sqrt{5} - 9\sqrt{5}\), both terms share the \(\sqrt{5}\). To combine them, we only need to focus on the coefficients (the numbers in front of the square roots). The coefficients are 12 and -9. We perform the subtraction: 12 minus 9 equals 3. So the combined term becomes \(3\sqrt{5}\). Combining like terms simplifies the expression and makes it easier to work with or use in further calculations.

The concept of a coefficient is important when simplifying algebraic expressions. A coefficient is a number that is multiplied by a variable or a square root. In an expression like \(4\sqrt{45}\), the number 4 is the coefficient. It indicates how many times the square root is taken. In the process of simplifying, we often need to rearrange or adjust these coefficients when combining like terms. For instance, in \(4 \times 3\sqrt{5}\), we multiply the coefficients 4 and 3 from \(\sqrt{45}\) and \(3\sqrt{5}\) to get a new coefficient of 12. Understanding coefficients helps in simplifying and accurately solving algebraic expressions, ensuring that the expressions remain mathematically correct when terms are combined.