Problem 63 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable. $$ (3 x)^{2} \text { when } x=4 $$

Step-by-Step Solution

Answer

The evaluation of the expression \( (3 x)^{2} \) where \(x=4\) is 144.

Step 1: Substitute the Value of the Variable

Replace the variable \(x\) with the provided value. This replaces \(3x\) in the expression with \(3*4\), resulting in \( (3*4)^{2}\).

Step 2: Apply Operator Precedence

According to operator precedence, multiplication comes before the power operation, so we first calculate \(3*4\) which results in 12. Now, the expression becomes \(12^{2}\).

Step 3: Apply the Power Rule

The power rule means one should multiply the base by itself for the number of times equal to the exponent. Hence, \(12^{2}\) results in \(12*12\) which equals 144.

Key Concepts

Substitution in Algebra

Substitution in algebra is when you replace variables with numbers in an expression or equation. It simplifies the problem and allows you to compute a specific value.
When tackling an algebraic expression, like \((3x)^2\) with \(x = 4\), the first step is to substitute 4 for every instance of \(x\).
This transforms the expression to \((3 \times 4)^2\).

• Identify the variable in the equation.
• Replace it with the given number every time it appears.
• Proceed to solve the now numerical expression.

This step is crucial since it sets the foundation for all subsequent calculations.

Order of Operations

The order of operations is essential in solving algebraic expressions correctly. It dictates the sequence in which different parts of a mathematical expression should be calculated.
This sequence is often remembered using the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). In our example, \((3 \times 4)^2\),
we must multiply \(3\) by \(4\) before dealing with the exponent, because multiplication comes before exponentiation.

• Always perform calculations inside parentheses first.
• Next, handle any exponents.
• Proceed with multiplication and division from left to right.
• Finish up with addition and subtraction from left to right.

Completing steps in the right order ensures the expression's value is accurate.

Exponentiation

Exponentiation is a mathematical operation involving powers. In simple terms, it means multiplying a number by itself a certain number of times. For instance, the expression \(12^2\) means you multiply 12 by itself once.
In our solution, once we have \(12^2\), the value is solved by finding: \[12 \, \text{times} \, 12 = 144\]

• Exponents show how many times to multiply the base by itself.
• An exponent of 2 is often read as "squared" and relates to the area of a square.
• Exponentiation is a shortcut to repeated multiplication.

Understanding how exponents work simplifies your ability to evaluate expressions quickly.

Algebraic Expressions

Algebraic expressions combine numbers, variables, and arithmetic operations like addition and multiplication. They may look complex, but they are broken down through systematic steps.
In our scenario, \((3x)^2\) is an algebraic expression involving both a variable and an exponent.
Understanding its components lets you rearrange it into a numerical expression after substitution, which then follows the order of operations.

• Variables: Symbols used to represent numbers.
• Coefficients: Numbers that multiply the variables.
• Operations: Addition, subtraction, multiplications, and other math functions.

Algebraic expressions are foundational in mathematics for forming equations and developing formulas.