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One of the key steps in working with algebraic expressions is substituting variables with their specific numerical values. This process involves identifying all instances of a variable within an expression and replacing them with a given number. It’s crucial to substitute variables carefully to avoid errors. For example, if an expression includes variables like x and z, and we know that x = 8 and z = 12, we must replace every x with 8 and every z with 12 before proceeding with any further calculations. This substitution lays the groundwork for correctly evaluating the entire expression.

Exponents represent a shorthand way to indicate the multiplication of a number by itself a certain number of times. The expression a^n, where n is a whole number, tells us to multiply the base a by itself n times. For instance, 3^2 means 3 multiplied by 3, or 9. In the context of our example, 8^2 would be 8 times 8, equaling 64.

Understanding how to work with exponents is essential for simplifying expressions. Exponents follow specific rules, such as the power of a product and power of a power that must be applied correctly to simplify expressions correctly.

Simplifying expressions is the process of breaking down complex algebraic expressions into their simplest form. After substituting variables and handling exponents, the remaining arithmetic operations like addition, subtraction, multiplication, and division are performed in accordance with the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

By simplifying an expression, you're seeking to make it as straightforward as possible, often with the goal of isolating a variable or calculating a numerical value. In our sample problem, after substituting variables and calculating the exponent, the expression 8^2 + 12 simplifies to 76.

Arithmetic operations are the foundation of all mathematical computations. These include addition, subtraction, multiplication, and division. Once variables have been substituted and exponents have been calculated, these operations are carried out to solve an expression or equation.

In our sample problem, after computing the exponent 8^2, which equals 64, the last step is to perform the arithmetic operation of addition by combining this result with 12 to obtain the final answer of 76. This step is straightforward, but it's important to approach it methodically to ensure accuracy in the solution.