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Chapter 1: Problem 63

Rewrite each expression using the distributive property. Simplify if possible. $$-5(z+3)$$

Short Answer

Expert verified
The simplified expression using the distributive property is: \(-5(z + 3) = -5z - 15\).

Step by step solution

01

Apply the Distributive Property

To apply the distributive property, we have to multiply each term inside the parentheses by the number outside the parentheses, which is -5. So, we need to multiply -5 by both z and 3.
02

Multiply -5 by z

To multiply -5 by z, simply write them together: \(-5 \cdot z = -5z\).
03

Multiply -5 by 3

To multiply -5 by 3, perform the multiplication: \(-5 \cdot 3 = -15\).
04

Rewrite the expression

After applying the distributive property, we can rewrite the expression as: \(-5(z + 3) = -5z - 15\). The simplified expression using the distributive property is: \[ -5(z + 3) = -5z - 15 \]

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Key Concepts

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

Algebraic Expressions
Algebraic expressions can often seem daunting at first. They are simply a combination of numbers, variables, and operations. An expression like \(-5(z+3)\)is a typical algebraic expression that includes multiplication and addition. Here,
  • \(-5\) is a number, also known as a coefficient,
  • \(z\) is a variable representing an unknown value,
  • \(3\) is a constant.

These elements are combined using mathematical operations like addition and multiplication to represent a particular quantity. Understanding how to manipulate these expressions is key to solving algebraic problems. The distributive property is one such tool that helps us manage and simplify them.
Simplifying Expressions
Simplifying expressions is an essential skill in algebra to make complex expressions easier to work with. When you simplify an expression, you perform various operations to get a more streamlined version. Using the distributive property is a common way to simplify expressions like \(-5(z+3)\).Here's how:
  • Distribute -5 across \((z + 3)\)to get \(-5z\) and \(-5 \cdot 3\).
  • Simplify \(-5 \cdot 3\)to get \(-15\).
  • Combine terms to write the expression as \(-5z - 15\).

After simplifying, the expression might look less complex and can be more easily evaluated or solved, depending on the context. The goal is to rewrite the expression without changing its value.
Multiplication in Algebra
Multiplication in algebra is similar to basic arithmetic, but it can involve variables and expressions. Multiplying a number by an expression requires spreading the number across terms within parentheses. For example, in \(-5(z+3)\),you multiply \(-5\)by each term inside the parenthesis.
To do this:
  • Multiply \(-5\)by \(z\)to get \(-5z\).
  • Multiply \(-5\)by \(3\)to get \(-15\).

This operation is part of the distributive property, which is crucial for expanding expressions and simplifying equations in algebra. Practicing these multiplications helps solidify your understanding of how numbers and variables interact in algebraic formulas.

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Most popular questions from this chapter

Write an expression for each and simplify. Answer the question with a complete sentence. In \(2002,\) the total attendance at Major League Baseball games was \(67,859,176 .\) In \(2003,\) that figure was \(67,630,052 .\) What was the difference in the number of people who went to ballparks from 2002 to \(2003 ?\)

Rewrite each expression using the distributive property. Simplify if possible. $$-2(m+11)$$

Write a mathematical expression for each and simplify. \(-25\) less than \(-19\).

Use the order of operations to simplify. $$(9-14)^{2}-(-3)(6)$$

Applications of Perimeter, Area, and Volume: Use 3.14 for \(\pi\) and include the correct units. Phil plans to put a fence around his \(9 \mathrm{ft} \times 5 \mathrm{ft}\) rectangular garden to keep out the rabbits. The fencing costs \(\$ 1.60 / \mathrm{ft}\). How much will the fence cost?
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