91Ó°ÊÓ

Substitution is a foundational method in algebra that involves replacing variables with actual numbers to evaluate expressions. Think of it as a 'search and replace' function in a word editor, but for math equations. When you are given specific values for the variables, you plug them into the equation in place of the variables.

For example, if an equation includes variables like \( x, y, \) and \( z \), and you're told that \( x=-2, y=5, \) and \( z=-3, \) you would replace every instance of \( x \) with -2, \( y \) with 5, and \( z \) with -3. This process transforms an abstract expression into a concrete numerical one that is ready for simplification. It's crucial to substitute carefully and accurately to avoid errors that can throw off the entire solution.

Simplifying expressions is a lot like condensing a long sentence into a shorter one without changing its meaning. In math, this means merging terms into their simplest form to make calculations easier and expressions neater. After substitution, you'll often find that the expression can be simplified by carrying out operations like multiplication, division, addition, and subtraction.

In our problem, once the values of \( x, y, \) and \( z \) are substituted, we obtain numerical terms next to each other. They can be multiplied or divided directly. It's essential to follow the correct order of operations, often remembered by the acronym PEMDAS (parentheses, exponents, multiplication and division from left to right, addition and subtraction from left to right). By simplifying step by step, we ensure the expression is reduced to the simplest numerical value possible.

Exponents can be seen as shorthand for repeated multiplication. The rules governing them help us manage multiplication of powers and the simplification of expressions containing exponents. Let's cover two basic rules that apply in our example:

The Product of Powers Rule

• It states that when multiplying two powers that have the same base, you add the exponents. For example, \( a^m \times a^n = a^{m+n} \).

The Power of a Power Rule

• This rule tells us that when you raise a power to another power, you multiply the exponents. For instance, \( (a^m)^n = a^{m \times n} \).

In the exercise, we encounter \( (-3)^2 \). According to the exponent rules, this means \( -3 \) is multiplied by itself twice, giving us 9. Mastering these exponent rules is an invaluable skill in algebra as it simplifies complex expressions into manageable pieces.