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Chapter 1: Problem 10

Simplify each expression. \(8(2 p-9 q)\)

Short Answer

Expert verified
16p - 72q

Step by step solution

01

Distribute the Constant

First, distribute the constant outside the parentheses to each term inside the parentheses. In this case, distribute 8 to both terms in the expression inside the parentheses: \[ 8(2p - 9q) \]
02

Multiply the Constant with Each Term

Multiply 8 by each term inside the parentheses: \[ 8 \times 2p = 16p \] and \[ 8 \times -9q = -72q \].
03

Combine the Results

After distributing the constant, combine the results together to get the simplified expression: \[ 16p - 72q \]

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Key Concepts

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

Distribution in Algebra
Distribution in algebra is a technique where we multiply a single term (often called a constant or a coefficient) by each term inside a set of parentheses.
This method helps to simplify expressions and is also known as the distributive property.
In our exercise, we used distribution to simplify the expression \(8(2p - 9q)\).

The process involves:
  • Taking the constant outside the parentheses (8 in this case).
  • Multiplying this constant by each term within the parentheses (\(2p\) and \(-9q\)).
Using the distributive property makes it easier to handle and break down complex expressions into simpler, more manageable parts.
Constant Multiplication
Constant multiplication means multiplying a constant term by another term or expression.
It is fundamental in algebra and appears frequently in problems involving distribution.
In our exercise, we multiplied the constant 8 by the terms inside the parentheses.

This process involves:
  • Identifying the constant and the terms it needs to be multiplied with.
  • Performing the multiplication for each term.
  • Ensuring the signs are correctly applied (positive or negative).
For example, in our exercise:
  • The term \(8 \times 2p\) becomes \(16p\).
  • The term \(8 \times -9q\) becomes \(-72q\).
Correct constant multiplication ensures that expressions are simplified accurately.
Combining Like Terms
After distributing and multiplying constants, the next step is to combine like terms.
Like terms have the same variable raised to the same power, and their coefficients can be added or subtracted.
This step makes the expression simpler and easier to work with.

In our exercise, we didn't have like terms to combine, but the final expression after distribution and multiplication was: \(16p - 72q\).
Here's the basic process for combining like terms:
  • Identify terms with the same variables and exponents.
  • Add or subtract their coefficients.
  • Simplify the expression as much as possible.
Combining like terms helps to consolidate terms in an algebraic expression, reducing it to its simplest form.

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