Problem 10 - Algebra 1: Concepts and Skills

Question

Evaluate the variable expression when t = 3. \(1+t^{3}\)

Step-by-Step Solution

Answer

Evaluating the variable expression \(1+t^{3}\) when t = 3 gives us a result of 28.

Step 1: Substitution of the variable t

The variable t in the expression \(1+t^{3}\) is replaced with its given value, which is 3. This results in an expression \(1+ (3)^{3}\).

Step 2: Calculating the exponent

We calculate the value of \(3^{3}\) which results in 27. So we update the above expression make it \(1+ 27\).

Step 3: Addition of the numbers

The final step is to add 1 and 27 together to get the final result. This results in a value of 28.

Key Concepts

Variable Evaluation

Variable evaluation is a crucial concept in algebra where you determine the numerical value of expressions based on specific values assigned to variables. In simpler terms, it means "figuring out what happens when we plug in a number for a letter." This is extremely useful because it allows us to work with symbolic representations and see how they behave with different numbers.

For example, in the expression given: \(1 + t^{3}\), we have the letter \(t\) acting as a variable. In this exercise, we are instructed to evaluate the expression by assuming \(t = 3\). This means we're calculating the expression when the letter \(t\) is replaced by the number 3. This step is called "substitution."

It’s important to accurately evaluate the variable by following these steps:

• Identify the variable in the expression.
• Replace or substitute the variable with the given numerical value.
• Simplify the expression to find the value.

This allows us to convert an algebraic expression into a simpler arithmetic problem.

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. It allows us to write repeated multiplication compactly and is commonly used in algebraic expressions to denote powers of numbers.

In our exercise, the expression \(t^3\) requires us to raise the number 3 to the power of 3. The base here is 3, and the exponent is also 3. This means that 3 is multiplied by itself a total of 3 times, which can be written as:

\[3^3 = 3 \times 3 \times 3 = 27\]

Learning to perform exponentiation is important because it simplifies the multiplication of similar numbers.

• The base is the number you are multiplying.
• The exponent tells you how many times to multiply the base by itself.
• Calculate it step-by-step if necessary to avoid errors.

This will help ensure accuracy, especially when working with larger numbers or more complex expressions.

Substitution in Expressions

Substitution is a method in algebra used to replace variables with their actual values or with other expressions. It plays a key role in simplifying expressions, solving equations, and evaluating functions.

When performing substitution, the main goal is to replace variables with known values and simplify the expression step-by-step. Let's see how this works using the expression \(1 + t^{3}\).

First, identify what value or expression you need to substitute for \(t\). In our example, \(t = 3\). Substituting \(t\) with 3 turns the expression into \(1 + 3^3\).

Next, compute the substituted parts first—in this case, the exponentiation—in \(1 + 3^3\), **3 multiplied by itself 3 times equals 27.** Then, perform any remaining arithmetic operations, such as addition in this example. Adding 1 to 27 gives the final result of 28.

• Always substitute variables with care to avoid mistakes.
• Simplify the expression methodically after substitution.
• Follow operations rules: handling exponents before multiplication, addition, or subtraction.

This understanding of substitution aids in managing expressions and contributes to solving more complicated algebraic problems.