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Chapter 1: Problem 40

Rewrite each expression using the indicated property. \(-8(c+5) ;\) distributive

Short Answer

Expert verified
The rewritten expression using the distributive property is: \(-8(c+5) = -8c - 40\).

Step by step solution

01

Identify the terms that need to be distributed

In the given expression \(-8(c+5)\), the distributive property needs to be applied to the terms -8 and (c+5). In this case, -8 is the "a" value and (c+5) corresponds to (b+c).
02

Apply the distributive property

Using the distributive property formula (a(b+c) = ab + ac), we can rewrite the expression as follows: \(-8(c+5) = -8c + (-8)(5)\)
03

Perform the multiplication

Now, we can multiply -8 by c and -8 by 5: \(-8c + (-8)(5) = -8c - 40\) So, the rewritten expression using the distributive property is: \(-8(c+5) = -8c - 40\)

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Key Concepts

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

Algebraic Expressions
Algebraic expressions are a fundamental part of algebra, serving as a means to represent relationships and computations. An algebraic expression is composed of numbers, variables (like \(c\)), and arithmetic operations. These components are combined to form expressions such as \(-8(c+5)\). Understanding how to manipulate these expressions is crucial for solving algebraic equations and simplifying problems.

When working with algebraic expressions:
  • Recognize that they can represent a variety of mathematical ideas.
  • Realize they typically consist of terms, which are parts of an expression separated by plus or minus signs. For example, \(-8c\) and \(-40\) are terms.
  • Understand that expressions can be simplified or rewritten using properties like the distributive property.
Such skills in handling algebraic expressions don't just help in solving equations but also in understanding word problems and real-world scenarios.
Multiplication
Multiplication is a basic arithmetic operation that is frequently used in algebra. It involves scaling one quantity by another, which in algebra often manifests in the form of multiplying coefficients with variables or numbers within an expression. In our example \(-8(c + 5)\), multiplication happens when we distribute the \(-8\) across the terms inside the parentheses.

Key points about multiplication in algebra include:
  • The numbers or variables being multiplied are called factors, and the result is a product.
  • When multiplying terms, we separately multiply the coefficients and the variables. For instance, multiplying \(-8\) and \(c\) gives the product \(-8c\).
  • It helps to remember that multiplying by a negative number affects the sign of the product, as shown when \(-8\) is multiplied by \(+5\) to yield \(-40\).
Understanding multiplication's role in algebraic manipulations is essential for applying properties like the distributive property effectively.
Rewrite Expressions
Rewriting expressions is an essential skill in simplifying and solving equations effectively. It often involves applying properties of arithmetic operations, like the distributive property, to transform the expression into an equivalent but possibly more useful or simpler form.

To rewrite expressions like \(-8(c+5)\):
  • Identify which algebraic properties can be applied. Here, the distributive property is key.
  • Apply these properties to break down and simplify the original expression. Distribute \(-8\) to both \(c\) and \(+5\), resulting in \(-8c - 40\).
  • This process helps in revealing the internal structure of the expression, making it easier to handle in larger equations or word problems.
Simplifying or rewriting doesn't change the expression's value, just its form, facilitating easier computation or comparison.

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Most popular questions from this chapter

Rewrite each expression using the distributive property. Simplify if possible. $$-(v-6)$$

Divide. $$\frac{-84}{-210}$$

Rewrite each expression using the distributive property. Simplify if possible. $$(8-2) 4$$

Objectives 3 and 4 Clarice's parents tell her that she must deposit \(\frac{1}{3}\) of the money she earns from babysitting into her savings account, but she can keep the rest. If she earns \(\$ 117\) in 1 week during the summer how much does she deposit, and how much does she keep?

Write a mathematical expression for each and simplify. 18 less than the sum of \(-5\) and 11.
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