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Chapter 2: Problem 64

Write each phrase as an algebraic expression. The product of \(-3\) and \(z\)

Short Answer

Expert verified
-3z

Step by step solution

01

Identify the mathematical operation

The word 'product' indicates that we need to multiply two numbers or variables. In this case, we are multiplying \(-3\) and \(z\).
02

Set up the algebraic expression

To create the algebraic expression for the given phrase, place \(-3\) and \(z\) together with a multiplication symbol in between. This gives us \(-3z\).
03

Simplify the expression

In algebra, it is common practice to omit the multiplication symbol when multiplying a number by a variable. Therefore, the simplified algebraic expression is \(-3z\).

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Key Concepts

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

Product
In algebra, the term 'product' signifies a multiplication operation. It's essential to recognize this terminology since it guides us to the appropriate mathematical action. For example, the exercise asks for the product of \(-3\) and \(z\). When we see 'product', we immediately know we need to multiply the values. Understanding such terms helps in translating word problems into algebraic expressions.

Some common words that indicate a product include:
  • Times
  • Multiplied by
  • Of

So, when seeing 'the product of two values', it simply directs us to use the multiplication operation.
Multiplication
Multiplication in algebra involves combining numbers or variables to form a single term. In the exercise, we multiply \(-3\) and \(z\). This means combining them into one product, written as \(-3z\).

Multiplication can involve:
  • Two numbers (e.g., 2 \times\ 3 = 6)
  • A number and a variable (e.g., \(-3z\))
  • Two variables (e.g., \(xy\))

Here, we don't need to write the multiplication symbol explicitly when a number and a variable are multiplied. This is an algebraic convention for simplification and clarity.
Simplification
Simplifying an algebraic expression makes it easier to read and work with. In this exercise, simplification refers to writing \(-3 \times z\) as \(-3z\). This is because multiplication between a number and a variable is implied in algebra.

Steps for simplification include:
  • Removing unnecessary symbols
  • Combining like terms
  • Reducing fractions

Simplification helps prevent confusion and keeps expressions tidy. While \(-3 \times z\) and \(-3z\) mean the same thing mathematically, the latter is the preferred, simplified form.
Variables
Variables are symbols used to represent unknown values or quantities in algebra. They can be any letter, like \(x\), \(y\), or in our case, \(z\).

Key points about variables:
  • They can represent any number
  • They allow for generalizations in math problems
  • They help in formulating equations and expressions

In the given exercise, \(z\) is the variable, and \(-3\) is the coefficient. When we say the product of \(-3\) and \(z\), we are forming an expression where \(z\) is multiplied by \(-3\), leading to the expression \(-3z\). This results in an algebraic representation that can be used for further calculations or problem-solving.

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Most popular questions from this chapter

Assume \(a>0\) (this means that \(a\) is positive) and \(b<0\) (this means that \(b\) is negative). Find the sign of each expression. $$-|a|$$

Translate each English phrase to a mathematical expression. Then simplify. The difference of 105 and 110

Perform the indicated operation. $$\frac{0}{-2}$$

Assume \(a>0\) (this means that \(a\) is positive) and \(b<0\) (this means that \(b\) is negative). Find the sign of each expression. $$|a|+|b|$$

Simplify. $$(-4)^{4}$$
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