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Chapter 7: Problem 136

In the following exercises, simplify using the distributive property. $$9(8 x-3)-(-2)$$

Short Answer

Expert verified
\(72x - 25\)

Step by step solution

01

Apply the distributive property

Apply the distributive property to the term inside the parentheses: \(9(8x - 3)\). This means multiplying 9 by each term inside the parentheses separately. \[9 \times 8x - 9 \times 3\]
02

Simplify the expression

Perform the multiplication and simplify: \[9 \times 8x = 72x\] and \[9 \times 3 = 27\]. So the expression becomes: \[72x - 27\].
03

Address the subtraction

Subtracting a negative number is the same as adding the opposite. Therefore, change \(-(-2)\) to \(+2\). The overall expression is now: \[72x - 27 + 2\].
04

Combine like terms

Combine the constant terms: \[-27 + 2 = -25\]. This simplifies the expression to: \[72x - 25\].

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Key Concepts

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

Simplifying Expressions
Simplifying expressions is an important skill in algebra. It involves rewriting an expression in a simpler form without changing its value.

When simplifying, always perform operations such as addition, subtraction, multiplication, and division.

In our example:
  • We started with \(9(8x - 3) - (-2)\).
  • We then applied distributive property. More on this in the next section.
  • After simplifying further, we reached the final result \(72x - 25\).
    • Simplifying algebraic expressions is a process that brings clarity and makes the overall work more manageable.
Combining Like Terms
In algebra, combining like terms makes an expression simpler to work with.

Like terms are terms that contain the same variable raised to the same power.
For example, in the expression \(5x + 3 + 2x - 7\), both \(5x\) and \(2x\) are like terms.
  • To combine them, simply add their coefficients: \(5x + 2x = 7x\)
  • Combine also the constants: \(3 - 7 = -4\)
In our example, combining like terms happens in the step:
\(72x - 27 + 2\)
where we combine the constants \(-27\)and \(+2\) to get \(-25\).
Combining like terms concisely aggregates similar components, making the expression more straightforward and easier to handle.
Applying Distributive Property
The distributive property is a useful tool in algebra that lets you simplify expressions by distributing a number across terms inside parentheses. The rule is:

\(a(b + c) = ab + ac\)

Our example: \(9(8x - 3) - (-2)\) shows this in action.
  • First, distribute \(9\) to both \(8x\) and \(-3\).
  • Thus, \(9(8x) = 72x\) and \(9(-3) = -27\).
  • The expression becomes \(72x - 27\).
Remember, the distributive property is particularly useful when simplifying expressions with parentheses. It sets the stage for further steps, including combining like terms and finalizing the simplified form of the expression.

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