Problem 33 - Beginning and Intermediate Algebra

Question

Use the distributive property to write each expression without parentheses. Then simplify the result. See Example 4. \(9(x-6)\)

Step-by-Step Solution

Answer

The simplified expression is \(9x - 54\).

Step 1: Apply the Distributive Property

The distributive property states that for an expression in the form of \(a(b + c)\), it can be expanded as \(ab + ac\). Apply this property to the expression \(9(x - 6)\). Multiply 9 by each term inside the parentheses: \(9 \times x\) and \(9 \times (-6)\).

Step 2: Perform the Multiplication

Calculate the products from Step 1. For \(9 \times x\), the result is \(9x\). For \(9 \times (-6)\), the result is \(-54\). So, the expression becomes \(9x - 54\).

Key Concepts

Expression Simplification

Expression simplification in algebra involves reducing an expression to its simplest form. This helps in understanding and solving problems more easily. It is about breaking down complex mathematical expressions into more manageable components.
Take for instance the expression we've dealt with:

• You start with something like \(9(x - 6)\).
• Upon removing the parentheses using the distributive property, you obtain \(9x - 54\).

The simplest form of this expression is when no further direct calculations can reduce it anymore.
Nonetheless, checking for common factors across the terms provides a way to simplify even further when possible.
Understanding expression simplification not only aids in solving but also in understanding the elegance and order of algebraic expressions.

Algebraic Multiplication

Algebraic multiplication is a crucial part of simplifying expressions, especially when applying properties like the distributive property. It's not just about multiplying numbers, but understanding how to manage terms and variables.
In the given solution, we begin with \(9(x - 6)\):

• The task initially is to multiply each term inside the parentheses by 9.
• I.e., \(9 \times x = 9x\) and \(9 \times (-6) = -54\).

After executing these operations, the expression expands into \(9x - 54\).
This technique turns more complex expressions into easier, standalone terms, setting the stage for further manipulations or simplifications if needed.
Mastering algebraic multiplication enables you to tackle a vast array of algebra problems with confidence and precision.

Parentheses Removal

Removing parentheses in algebra simplifies expressions and reveals their inner structure. This is initially done through properties like the distributive property.
In our example, the expression \(9(x - 6)\) required using the property to eliminate parentheses:

• Start by distributing \(9\) over the terms, thus effectively removing the grouping symbol \((x - 6)\).
• The expanded form \(9x - 54\) shows the outcome of this operation.

Parentheses often indicate that terms inside are to be multiplied or operated on before dealing with others outside.
Skills in removing parentheses are foundational; they ensure that expressions are handled correctly and calculations proceed in the right order.
Grasping the role of parentheses can greatly enhance your accuracy when working with more complex mathematical scenarios.