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Chapter 7: Problem 425

In the following exercises, simplify the given expression. $$ 8(6 p-1)+2(9 p+3) $$

Short Answer

Expert verified
The simplified expression is \( 66p - 2 \).

Step by step solution

01

Distribute the Constants

Apply the distributive property to remove the parentheses. Multiply the constants by each term inside the parentheses: \[ 8(6p - 1) + 2(9p + 3) \rightarrow 8 \cdot 6p + 8 \cdot (-1) + 2 \cdot 9p + 2 \cdot 3 \]
02

Simplify Each Term

Perform the multiplication for each term: \[ 8 \cdot 6p = 48p \] \[ 8 \cdot (-1) = -8 \] \[ 2 \cdot 9p = 18p \] \[ 2 \cdot 3 = 6 \] Now, the expression is: \[ 48p - 8 + 18p + 6 \]
03

Combine Like Terms

Combine the terms that contain the variable \( p \) and the constant terms separately: \[ 48p + 18p = 66p \] \[ -8 + 6 = -2 \] The simplified expression is: \[ 66p - 2 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
To simplify expressions like the one in the exercise, we often use the distributive property. The distributive property allows us to multiply a single term by each term inside a pair of parentheses. This is expressed mathematically as:

\[ a(b + c) = ab + ac \]
In the given exercise, \[ 8(6p-1) + 2(9p+3) \] you distribute the constants 8 and 2 to the terms inside the parentheses. This results in:
\[ 8 \times 6p + 8 \times (-1) + 2 \times 9p + 2 \times 3 \] Hence, the intermediate step can be converted into separate terms, simplifying our computation.
Combining Like Terms
Once we have used the distributive property and gotten rid of the parentheses, the next step is to combine like terms. Like terms are terms that contain the same variables raised to the same power.
For instance, in the expression from our example:
\[ 48p - 8 + 18p + 6 \]
The like terms are:
  • \[ 48p \] and \[ 18p\]
  • \[ -8 \] and \[ 6 \]

Combining them, we get:
\[ 48p + 18p = 66p \] and \[ -8 + 6 = -2 \]
So, the expression simplifies to:
\[ 66p - 2 \] This step is crucial as it reduces the complexity of the expression, making it simpler to work with in any following math problems.
Prealgebra Operations
Prealgebra lays the foundation for all future algebra work by introducing basic operations. Simplifying expressions is one such foundational skill.
  • Understanding and applying the distributive property
  • Combining like terms
  • Adding, subtracting, multiplying, and dividing constants and variables

Consider our original expression:
\[ 8(6p-1) + 2(9p+3) \]
By knowing how to distribute constants and combine like terms correctly, we:
  • First distribute constants inside the parentheses
  • Then combine any terms that are alike

Prealgebra is about mastering these techniques, which are essential when you encounter more complex algebraic equations. Practice these steps to build a solid foundation!

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