Problem 7 - Elementary and Intermediate Algebra

Question

a. Fill in the blanks to simplify the expression. \(4(9 t)=\) \((\quad \cdot \quad) t=\quad t\) b. What property did you use in part a?

Step-by-Step Solution

Answer

36; Associative Property of Multiplication.

Step 1: Apply Distribution

The expression given is \(4(9t)\). We use the associative property to rewrite this as \((4 \cdot 9) \cdot t\).

Step 2: Calculate the Coefficient

Now we multiply the numbers, \(4 \cdot 9 = 36\). So, the expression becomes \(36 \cdot t\).

Step 3: Write the Simplified Expression

After performing the calculations, the initial expression \(4(9t)\) simplifies to \(36t\).

Key Concepts

Associative Property

The associative property is a fundamental concept in algebra. It refers to the way in which numbers are grouped within an expression and asserts that the way these numbers are grouped does not change the result. This property applies to both addition and multiplication. However, it does not apply to subtraction or division.

Let’s illustrate with a clear example. Consider the multiplication expression \((a \cdot b) \cdot c\). According to the associative property, you can rearrange this as \(a \cdot (b \cdot c)\) without affecting the result. In the context of the original exercise, \(4(9t)\) can be seen first as \((4 \cdot 9) \cdot t\) or as \(4 \cdot (9 \cdot t)\). Either structure will lead to the same simplified product, \(36t\), demonstrating the associative nature of multiplication.

Understanding this property can significantly simplify complex algebraic expressions, making calculations easier and freeing you to rearrange groupings as needed for simplification.

Expression Simplification

Expression simplification involves reducing a mathematical expression to a simpler or more manageable form. In algebra, this often means performing operations and combining like terms to present the expression in a way that is easy to understand and work with.

Simplifying an expression like \(4(9t)\) involves a series of straightforward operations:

• First, apply the associative property, so you focus on simplifying within the parentheses: \((4 \cdot 9) \cdot t\).
• Next, perform the multiplication, calculating \(4 \cdot 9 = 36\).
• This transformation results in the simpler form of \(36t\).

Expression simplification is essential in algebra as it helps streamline your calculations and makes solving equations more efficient. It’s akin to cleaning up the workspace before starting on a project – removing unnecessary complexity allows you to better focus on the main task.

Algebraic Coefficients

Algebraic coefficients are the numerical factors that multiply the variables in an expression. Understanding them is crucial because they directly influence the value of the expression when the variable is replaced with a specific number.

In our exercise, the expression \(36t\) contains the coefficient 36, which emerged from simplifying the expression \(4(9t)\). Here’s how coefficients come into play:

• Coefficients can be multiplied or divided like regular numbers during simplification.
• They give scale to the variable – meaning that any value substituted for \(t\) will be multiplied by 36.
• Understanding coefficients helps in comparing or solving expressions, especially when equating different expressions or solving equations.

A deep understanding of algebraic coefficients is foundational for working with more complex algebraic equations, where you might need to solve for unknown variables by manipulating these coefficients.