Problem 3 - Algebra 1

Question

Evaluate the expressions \(3 x^{2}\) and \((3 x)^{2}\) when \(x=4\)

Step-by-Step Solution

Answer

The expression \(3 x^{2}\) evaluates to 48 and the expression \((3 x)^{2}\) evaluates to 144 when \(x=4\).

Step 1: Substitute x in \(3 x^{2}\)

First, we need to evaluate the expression \(3 x^{2}\). We substitute \(x=4\) into the expression, which gives us \(3 * 4^{2}\).

Step 2: Simplify expression \(3 x^{2}\)

Now we need to do the calculation. Following correct order of operations, we first calculate the exponent, \(4^{2} = 16\). Thus, the expression becomes \(3 * 16\). We then perform the multiplication, which gives us a result of 48.

Step 3: Substitute x in \((3 x)^{2}\)

Next, we substitute \(x=4\) into the expression \((3 x)^{2}\) which gives us \((3 * 4)^{2}\).

Step 4: Simplify expression \((3 x)^{2}\)

Now we need to do the calculation. This time, since the parentheses change the order of operations, we first perform the multiplication inside the parentheses: \(3 * 4 = 12\). Thus, the expression becomes \(12^{2}\). We then perform the exponentiation, which gives us a result of 144.

Key Concepts

Order of Operations

When solving mathematical expressions, it is crucial to follow the order of operations. This ensures accurate and consistent results. You may have heard of the acronym PEMDAS, which stands for:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In our example, we evaluated two expressions, and both required careful attention to this order.
The expression \(3 x^2\) required evaluating the exponent before multiplication. This means we calculated \(4^2\) to get 16, then multiplied by 3 to obtain 48.
On the other hand, the expression \((3x)^2\) involved parentheses, altering the order. We first multiplied inside the parentheses to get 12, followed by squaring it to achieve 144.

Exponents

Exponents are a way to express repeated multiplication of the same number. For example, \(x^2\) means \(x\) multiplied by itself.
In our problem, understanding exponents was key in simplifying the expressions effectively.
In \(3 x^2\), only \(x\) is raised to the power of 2. Hence, we calculated \(4^2\) to reach 16. Then, multiplying by 3 finalized our result to 48.

• In contrast, the expression \((3x)^2\) treated \(3x\) as a single entity.
• That is, \((3 * 4)^2\), allowing us to first compute inside the parentheses, and then square the result, which led to 144.

This distinction is vital in avoiding mistakes, particularly when variables are involved.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They allow us to represent real-world situations with mathematical symbols and solve problems.
In the given exercise, we worked with expressions that included both constants (like 3) and a variable \(x\). This variable was substituted with a specific value (4) to find the solution.

• Expression \(3 x^2\) combines the variable \(x\) and exponent to depict a square relationship.
• Expression \((3x)^2\) combines multiplication and exponentiation through the use of parentheses, highlighting compound operations.

Understanding how to evaluate and manipulate these expressions by correctly substituting variables and performing operations is essential in algebra.