Problem 26 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable. \(8 a\) when \(a=6\)

Step-by-Step Solution

Answer

The result of \((10-x)^{2}\) when \(x=6\) is 16

Step 1: Substitute the given value

First, the value for \(x\) given is 6. Substitute this \(x\) value into the expression to get: \((10-6)^{2}\)

Step 2: Carry out the subtraction

Perform the subtraction operation in the parentheses to simplify the expression: \(4^{2}\)

Step 3: Carry out the exponentiation

Exponentiation is the last operation to perform to find the value of the expression. Calculate \(4^{2}\) to obtain 16

Key Concepts

Substitution in Algebra

Understanding the process of substitution in algebra is crucial when we deal with algebraic expressions. Substitution is the act of replacing a variable in an expression with its numerical value. This is especially valuable when you're given a specific value for a variable, like in the exercise where we evaluate \( (10-x)^{2} \) for \( x=6 \).

Here's how substitution works. You start with an algebraic expression and replace each instance of the variable with the given number. In our example, \( x \) is replaced by 6, which transforms the expression into \( (10-6)^{2} \). This step simplifies the problem and prepares it for further operations. It's important to substitute carefully, ensuring that the correct value is placed and that the structure of the expression (like parentheses) is preserved.

Order of Operations

The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), tells us the sequence in which we should solve parts of a mathematical expression to ensure that everyone gets the same, correct result. After substitution, you apply the order of operations to solve the expression.

In our exercise, once we substitute 6 for \( x \), we get \( (10-6)^{2} \). According to PEMDAS, we first perform the operation inside the parentheses, simplifying \( 10-6 \) to 4, resulting in \( 4^{2} \). The subtraction comes before the exponentiation because it is inside the parentheses. This is a critical step, as mixing up the order can lead to incorrect answers.

Exponentiation

The concept of exponentiation refers to raising a number, known as the base, to the power of an exponent. The exponent indicates how many times the base is multiplied by itself. For instance, \( 4^{2} \) means that 4 is multiplied by itself once, resulting in 16.

Exponentiation is a powerful tool in algebra because it represents repeated multiplication concisely. In the context of our given exercise, after performing the subtraction and arriving at \( 4^{2} \) through substitution and applying the order of operations, we evaluate the exponent by multiplying 4 by itself, which equals 16. Understanding how exponentiation works enables you to handle more complex expressions involving powers and is vital in many areas of mathematics.