Problem 65 - Algebra 1

Question

Evaluate the expression for the given value of the variable. (Review 1.2) $$4\left(t^{3}\right) \text { when } t=3$$

Step-by-Step Solution

Answer

The value of the expression \(4t^3\) when \(t = 3\) is 108.

Step 1: Substitute the value of t

Firstly, substitute the value of the variable \(t\) in the expression with the provided value, which in this case is 3. The expression \(4t^3\) becomes \(4 \cdot (3)^3\).

Step 2: Compute the exponentiation

Next, perform the operation of exponentiation before multiplication as according to the BODMAS/BIDMAS rule, which states that operations in an expression are carried out in the following order: Brackets, Orders (that is, powers and square roots, etc.), Division, Multiplication, Addition and Subtraction. So, calculate \((3)^3\) (which means 3 times 3 times 3), giving a result of 27.

Step 3: Perform the multiplication

Finally, multiply the result from the previous step by 4 as indicated in the original expression. So, 4 times 27 equals 108.

Key Concepts

Exponentiation

Exponentiation is an essential mathematical operation that involves raising a number, known as the base, to the power of an exponent. In this particular exercise, the expression contains the term \( t^3 \). Here, 3 is the exponent that tells us how many times the base, \( t \), should be multiplied by itself. When \( t = 3 \), exponentiation translates to \( 3^3 \), which means multiplying 3 by itself twice more:

• First multiplication: \( 3 imes 3 = 9 \)
• Second multiplication: \( 9 imes 3 = 27 \)

The result of this exponentiation is 27. Understanding exponentiation is fundamental as it forms the basis for various algebraic operations, including those involving polynomials and scientific notation.
It's important to always perform exponentiation before moving on to multiplication or division when evaluating an expression.

Substitution

Substitution is a process used in algebra where you replace a variable in an expression with its given numerical value. It's a critical step in solving algebraic problems because it allows you to handle specific values directly. In our example, the expression \( 4t^3 \) needs to be evaluated for \( t = 3 \).
To do this, replace every instance of \( t \) in the expression with 3:

• Original expression: \( 4t^3 \)
• After substitution: \( 4 imes (3)^3 \)

Substitution simplifies the expression into a form that you can solve numerically. It's crucial to substitute accurately to avoid introducing errors into your calculations.
Make sure that you carefully follow any orders or operations after substituting, as this maintains the integrity of the math problem.

Order of Operations

The Order of Operations is a set of rules that indicate the sequence in which operations should be performed in a mathematical expression. This ensures that everyone arrives at the same answer when evaluating an expression. The acronym BODMAS/BIDMAS helps to remember this order: Brackets, Orders (exponents), Division and Multiplication (from left to right), Addition and Subtraction (from left to right).
In the context of our exercise, the expression \( 4 imes (3)^3 \) follows these steps:

• First, deal with the exponentiation: Compute \( (3)^3 = 27 \).
• Then, proceed with multiplication: Evaluate \( 4 imes 27 = 108 \).

Following the correct order of operations avoids misunderstandings and achieves the right results. Omitting this order will lead to incorrect answers, as different operations change the value of an expression. Hence, always ensure to apply these rules whenever dealing with mathematical expressions.