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Chapter 7: Problem 337

In the following exercises, simplity using the distributive property. $$ -8(-7 a-12) $$

Short Answer

Expert verified
56a + 96

Step by step solution

01

- Apply the Distributive Property

Use the distributive property to multiply \( -8 \) by each term inside the parentheses. The distributive property states that \( a(b + c) = ab + ac \). So, applying it here, \( -8(-7a - 12) = -8 \cdot (-7a) + (-8) \cdot (-12) \).
02

- Multiply the Constants with the Terms

Now multiply each term: \( -8 \cdot (-7a) = 56a \) and \( -8 \cdot (-12) = 96 \).
03

- Combine the Results

Combine the results of the multiplication: \( 56a + 96 \). This is the simplified form.

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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 a fundamental concept in algebra. It involves breaking down a complex expression into a simpler form, making it easier to understand and work with.
When you simplify an expression, you're typically combining like terms and performing basic operations to rewrite the expression in a more manageable form.

In the given exercise, the expression is simplified using the distributive property. By distributing and then combining the results, we arrive at the final simplified form. Simplifying expressions helps to solve mathematical problems more efficiently.
Prealgebra
Prealgebra is an introductory level of algebra that prepares students for more advanced algebra courses. It typically includes concepts such as:
  • basic arithmetic operations
  • working with variables
  • understanding the properties of numbers

In the context of prealgebra, understanding and applying the distributive property is crucial. This property allows you to multiply a single term by each term inside a set of parentheses, simplifying expressions and solving equations effectively.

Through the exercise, students practice using the distributive property in a prealgebra context, laying the groundwork for more advanced algebraic concepts.
Multiplication of Terms
The multiplication of terms is a key operation in algebra. It involves multiplying constants and variables to simplify expressions or solve equations.
A term can consist of a constant, a variable, or a combination of both. When multiplying terms, it's essential to:
  • Apply the distributive property correctly
  • Multiply constants (numbers) together
  • Multiply variables together by following the rules of exponents

In the example, \( -8(-7a - 12) \), the distributive property is used to multiply \( -8 \) by \( -7a \) and \(-12\): \[ -8(-7a - 12) = -8 \cdot (-7a) + (-8) \cdot (-12) \] which results in \ 56a + 96 \.
Understanding how to multiply terms correctly allows you to simplify expressions and solve equations, making these operations fundamental in prealgebra and beyond.

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In the following exercises, use the associative property to rewrite the given expression. (13 \cdot 5) \cdot 2=______
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