Problem 44 - Algebra 1: Concepts and Skills

Question

USING EXPONENTS Evaluate the expression. \(x^{2}-3\) when \(x=4\)

Step-by-Step Solution

Answer

The evaluated expression \(x^{2}-3\) when \(x=4\) is 13

Step 1: Substitution

For the given expression \(x^{2}-3\), substitute the given value of \(x\) which is 4. So, \(x^{2}-3\) becomes \(4^{2}-3\)

Step 2: Application of exponent

Next, the formula should be simplified by applying the exponent first, due to order of operations (PEDMAS/BODMAS rule). Thus, \(4^{2}\) equals 16. So, the expression becomes 16 - 3

Step 3: Final computation

After applying the exponent, do the subtraction. Thus, 16 - 3 equals 13

Key Concepts

Order of Operations

To correctly evaluate mathematical expressions, it's essential to follow a standard set of rules known as the 'order of operations.' This is a sequence that determines which operations to perform first to solve an equation accurately. The acronym PEMDAS is a popular way to remember these rules: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

For example, when faced with an expression such as \(x^{2}-3\) where \(x=4\), it is critical to address any operations within parentheses first, followed by exponents, before moving on to multiplication, division, addition, or subtraction. By systematically applying PEMDAS, one can avoid common mistakes and ensure the expression is correctly simplified.

Exponentiation

Exponentiation is an operation involving two numbers – the base and the exponent. The base is the number being multiplied by itself, and the exponent, commonly written as a superscript, denotes how many times the base is used as a factor. The expression \(4^{2}\) means that 4 should be multiplied by itself once since repetition is implied; hence, the result is 16.

When evaluating exponents in algebraic expressions, it's vital to perform this step immediately after any operations inside parentheses since it precedes multiplication and division in the order of operations. Understanding exponentiation is crucial for solving more complex expressions and equations, as it appears frequently in various areas of mathematics.

Substitution in Algebra

Substitution is a fundamental algebraic principle that involves replacing a variable with a number or another expression. This technique is widely used to simplify expressions or solve equations. In the context of the given expression \(x^{2}-3\), the substitution requires inputting the value of 4 in place of \(x\).

After substitution, the algebraic expression becomes numerical and much simpler to compute. This step is often taken before applying the order of operations to ensure the expression reflects the specific values of its variables, making further simplification and evaluation possible. Effective substitution skills are vital for students to progress in algebra and apply mathematical concepts to real-world problems.