Problem 53 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given values of the variables. (Lesson 1.2 ) \(g-h^{2}\) when \(g=4\) and \(h=8\)

Step-by-Step Solution

Answer

The expression \(g-h^2\) evaluates to \(-60\) when \(g=4\) and \(h=8\).

Step 1: Substitute the values into the expression

Replace \(g\) and \(h\) in the expression \(g-h^2\) with their given values. This gives \(4-8^2\).

Step 2: Simplify the expression

Next, simplify \(8^2\), which is 64. This gives \(4-64\).

Step 3: Subtract

Now, subtract 64 from 4. This gives \(-60\).

Key Concepts

Substitution in Algebra

Substitution is a foundational skill in algebra allowing you to find the value of expressions with variables. It involves replacing the variables in an algebraic expression with their corresponding numerical values. Consider an algebraic expression like \(g-h^{2}\). When given that \(g=4\) and \(h=8\), you substitute these numbers into the expression.

After substitution, your initial expression \(g-h^{2}\) becomes \(4-8^{2}\). The key here is to ensure that every instance of a variable is replaced with the value it represents. Without substitution, the expression is abstract and cannot be simplified to a numeric value. With substitution, you turn an abstract expression into a concrete arithmetic problem that can be solved step by step, as seen in the solution provided.

It is important to carefully substitute the values correctly to avoid common mistakes such as missing negative signs or squaring the wrong term. Always double-check your substitutions to ensure accuracy before proceeding to the simplification stage.

Simplifying Expressions

Once you've completed the substitution process, the next step in evaluating an algebraic expression is to simplify it. Simplifying involves performing arithmetic operations such as addition, subtraction, multiplication, and exponentiation. In our example, after substituting the values into \(g-h^{2}\), we get \(4-8^{2}\).

The expression simplification phase requires attention to the order of operations, often abbreviated as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Based on PEMDAS, we first evaluate the exponentiation before the subtraction, turning \(8^2\) into 64 and then simplifying to \(4-64\).

Simplifying skillfully gives you a clearer path to reach the expression's simplest form—which in this case is a single number. Simplification isn't always about getting to a number though; sometimes it’s about rewriting the expression in the most basic terms possible. Making sure each step of simplification is performed correctly will lead to accurate final results.

Exponentiation in Algebra

Exponentiation is an operation that raises a number to the power of another number. It is represented as a small raised number known as the exponent. In the expression \(g-h^{2}\), after substitution, we must calculate \(8^{2}\).

The exponent here is 2, which tells us to multiply the base, 8, by itself: \(8 \times 8 = 64\). Failing to correctly apply the rules of exponentiation can lead to significant errors. Understanding exponentiation rules, such as 'any number to the power of two means the number is multiplied by itself', helps avoid these errors.

In algebra, you'll encounter exponents frequently, and they can be more complex, including negative exponents or fractional exponents, each with its own set of rules. For the given exercise, always remember that handling the exponent comes after substitution but before carrying out any addition or subtraction outlined by the order of operations.