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Simplifying algebraic expressions is a fundamental process in algebra that involves reducing expressions to their simplest form. An expression is considered 'simplified' when it has no parentheses, no like terms to combine, and all the coefficients are reduced to their lowest terms.

For example, when given the expression \( -3x - 9y \) multiplied by \( -6y \) through the distributive property, the initial multiplication yields \( 18xy + 54y^2 \). To 'simplify' this, you look to see if there are any like terms to combine (terms with the same variables raised to the same power), but in this case, there are none. So the simplified expression is \( 18xy + 54y^2 \), where the coefficients are already in their lowest terms and no further reduction is possible.

Throughout the simplification process, it’s important to carefully perform operations according to the order of operations, recall the rules for multiplying variables and exponents, and keep track of plus and minus signs to ensure the simplified expression is accurate.

Multiplying variables is another core concept in algebra where variables are symbols that represent numbers. When multiplying variables, you have to follow certain rules to ensure the result is correct. One key rule is that when multiplying two variables of the same kind, you add the exponents.

Here's an example: \( x^a \times x^b = x^{a+b} \). This principle is applied in the earlier problem when multiplying \( -9y \times -6y \), resulting in \( 54y^2 \). The exponents on y are added together (1 + 1 to give y to the power of 2), showing how multiplication alters exponents in algebraic expressions.

It is also important to multiply the coefficients (the numerical part of the terms), separately from the variables, and then combine the results. A solid understanding of how to multiply variables can greatly simplify complex algebraic expressions.

Algebraic notation is the system of symbols and the rules for using those symbols used in algebra to express mathematical concepts. It is essential that students understand this notation as it is the language through which algebraic ideas are communicated.

In our exercise, the notation \( -3x \) indicates the multiplication of \( -3 \) and \( x \), even though there is no explicit multiplication sign. Also, the exponent notation \( y^2 \) describes y multiplied by itself. The correct interpretation of algebraic notation is crucial; misreading \( -6y \) as something other than \( -6 \times y \) or misinterpreting \( y^2 \) can lead to incorrect solutions. By mastering algebraic notation, students can read, write, and interpret algebraic expressions and equations correctly, enabling them to tackle a wide range of problems.