Problem 12 - Algebra 1: Concepts and Skills

Question

Evaluate the variable expression when t = 3. \((4 t)^{2}\)

Step-by-Step Solution

Answer

The evaluated variable expression \((4 t)^{2}\) when \( t = 3 \) is 144.

Step 1: Substitute the value of the variable

Substitute the value of \( t = 3 \) into the variable expression. This gives us \((4 * 3)^{2}\).

Step 2: Perform the multiplication

Perform the multiplication before moving on to the exponent. So, we will first multiply 4 and 3, which gives us \((12)^{2}\).

Step 3: Apply the exponent

Finally, you apply the exponent to your result which results in \( (12)^{2} = 144 \).

Key Concepts

Substitution

Substitution is the process of replacing a variable within an expression with a specific number or value. This is a fundamental step in algebra, as it allows us to evaluate expressions and solve equations. For example, if an expression contains a variable like \( t \), and you are given that \( t = 3 \), you "substitute" 3 in place of \( t \).

In the exercise, the expression \( (4t)^2 \) becomes \( (4 \times 3)^2 \) after substitution because \( t \) is replaced by 3. This step is crucial because it transforms an algebraic expression into one that can be directly calculated. Remember, substitution only works when a variable is given a specific value; otherwise, the expression remains abstract.

Exponents

Exponents are a way to represent repeated multiplication of the same number. They are written as a small number (the exponent) to the upper right of a base number. For instance, in \( n^2 \), the "2" is the exponent and "n" is the base, meaning \( n \times n \). In simpler terms, it tells you how many times to multiply the base by itself.

In our exercise, after replacing \( t \) with 3, we derived \( (4 \times 3)^2 \), which simplifies to \( 12^2 \). This means we multiply 12 by itself: \( 12 \times 12 = 144 \). Understanding what an exponent does helps simplify and solve such expressions easily. Exponents follow specific rules, which often fit into the broader set of rules called the "order of operations."

Order of Operations

The order of operations is the set of rules that dictates the sequence in which operations should be performed in mathematical expressions. Failing to follow this order can lead to incorrect results. The most common guiding principle is PEMDAS, which stands for:

• P: Parentheses first
• E: Exponents (i.e., powers and square roots, etc.)
• MD: Multiplication and Division (left-to-right)
• AS: Addition and Subtraction (left-to-right)

When applied, these rules help you break down complex expressions step-by-step. For the expression \( (4 \times 3)^2 \), the order goes as follows:

• First, handle any operations inside parentheses: \( 4 \times 3 = 12 \)
• Next, take care of the exponents: \( 12^2 = 144 \)

By adhering to the order of operations, you ensure that each part of the expression is calculated in the correct sequence, leading to the right answer.