Problem 62 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

The value of the expression \(4t^2\) when \(t=3\) is 36.

Step 1: Substitution

First, substitute \(t = 3\) into the given equation \(4t^2\). This results in: \(4(3)^2\).

Step 2: Perform the Calculations

Next, perform the calculation. First calculate \(3^2\) to get 9, then multiply 4 by 9 to get 36.

Key Concepts

Substitution

Substitution is a fundamental skill in algebra and is typically one of the first steps in solving an equation or evaluating an expression. It involves replacing a variable in an expression with a specified value to simplify or solve that expression. In this exercise, we are given the expression \(4t^2\) and the value of \(t = 3\). By substituting \(3\) for \(t\), we alter the original expression to \(4(3)^2\).

Substitution is especially useful because it allows us to evaluate expressions at specific points, making it easier to understand their behavior and see the results visibly. Key points to remember include:

• Ensure you substitute every occurrence of the variable.
• Double-check that you're using the correct value for the variable.

After performing substitution, we can proceed with further simplifications or calculations, such as arithmetic operations or evaluating exponents.

Exponents

Exponents show how many times a number, known as the base, is multiplied by itself. In the expression \(4(3)^2\), \(3\) is the base, and it is raised to the power of 2, which is the exponent. This calculation is thus \(3 \times 3\), which equals 9.

Understanding exponents is crucial because they often appear in algebra and mathematics, representing compact ways to express repeated multiplication. Make sure to:

• Identify the base and exponent correctly before performing calculations.
• Calculate the exponent part of a complex expression before doing any other operations (follow the order of operations).

Once we've calculated \(3^2\), we can substitute this result back into our expression, simplifying further with arithmetic operations.

Arithmetic Operations

Arithmetic operations include basic calculations: addition, subtraction, multiplication, and division. In our exercise, we focus on multiplication. The expression \(4(3^2)\) requires us to multiply \(4\) by the result of \(3^2\), which we previously determined to be \(9\).

Thus, the arithmetic operation we perform is \(4 \times 9\), resulting in \(36\). Important reminders for arithmetic operations include:

• Perform multiplication and division before addition and subtraction according to the order of operations (PEMDAS/BODMAS).
• Double-check calculations to avoid simple errors.

This final step gives us the evaluated value of the original expression, providing a complete understanding of how substitution, exponents, and arithmetic operations work together in algebraic problems.