Problem 11 - Algebra 1

Question

Evaluate the expression when \(x=3\) \((2 x)^{3}\)

Step-by-Step Solution

Answer

The evaluated expression when \(x=3\) for \( (2x)^3 \) is \(216\).

Step 1: Substitution

Substitute \(x\) with \(3\) in the given expression, which gives \( (2*3)^3 \).

Step 2: Apply Order of Operations

First, perform the operation inside the parenthesis (which is multiplication in this case). The expression becomes \( 6^3 \).

Step 3: Calculate Exponent

Finally, calculate \( 6^3 \) which gives a result of \(216\).

Key Concepts

Order of Operations

Understanding the order of operations is crucial when evaluating algebraic expressions. It's like a set of traffic rules for math that dictates which procedures to follow so that everyone arrives at the same answer.

The order of operations can be remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This rule means you should solve whatever is in parentheses first, then exponents, followed by multiplication and division, and finally, tackle any addition or subtraction. This hierarchy controls the calculations to ensure consistent results.

In our exercise, \( (2x)^3 \), after substituting \(x\) with \(3\), we first handle the operation inside the parentheses, which is multiplication (\(2 \times 3\)), before dealing with the exponent (\(6^3\)).

Exponentiation

Exponentiation is an operation that involves raising a number, called the base, to the power of an exponent. The exponent, located superscript to the base, represents how many times the base is multiplied by itself.

For example, in \(6^3\), the number \(6\) is the base and \(3\) is the exponent, indicating that \(6\) should be multiplied by itself \(3\) times (\(6 \times 6 \times 6\)). It's essential to not confuse exponentiation with multiplication or addition; it's a distinct operation that requires repeated multiplication.

In the context of our exercise, we calculated \(6^3\) after simplifying what was inside the parentheses, which gave us \(216\), as \(6\) multiplied by itself three times equals \(216\).

Substitution in Algebra

Substitution is a method used in algebra to replace variables with their corresponding numerical values. This technique is perfectly exemplified in our example \( (2x)^3 \), where we are asked to evaluate the expression for \(x=3\).

By substituting \(3\) for \(x\), the expression becomes \( (2*3)^3 \), which we then evaluated using the order of operations. Substitution allows us to transform an abstract algebraic expression into a numeric one that we can calculate. It’s like swapping out a temporarily unknown item with something known, allowing for further operation.

This step not only simplifies the problem at hand but also is a fundamental skill in algebra, often used to solve equations and inequalities. Ensuring that the substitution is done correctly lays the groundwork for accurately solving the expression.