Note 7.3.1.
Think of this problem in terms of money: if you owe five different people two dollars each, then you are actually in debt by $10.
Section 7.3 Multiplying and Dividing Signed Numbers
- Multiplication
- Division
- Powers of Negative Numbers
Subsection 7.3.1 Multiplication
Multiplication of integers is really just a short cut for repeated addition. For example, 
Similarly, 
Because the sum of five \(-2\)’s added together is \(-10\text{,}\) the product \(5(-2)\) is equal to \(-10\) also.
It is also true that
\begin{equation*}
(-2)(5) = 10
\end{equation*}
because we can multiply two numbers in either order and get the same answer.
From these examples, the following rule seems reasonable.
Opposite Signs.
The product of two numbers with opposite signs is negative.
Example 7.3.2.
- \(\displaystyle (-3)(6) = -18\)
- \(\displaystyle 8(-4) = -32\)
- \(\displaystyle 9(-0.3) = -2.7\)
- \(\displaystyle \left(-\dfrac{1}{2}\right)(6) = -3\)
Checkpoint 7.3.3.
Multiply.
- \(\displaystyle 5(-4)\)
- \(\displaystyle (-7)(2)\)
Answer.
- \(\displaystyle -20\)
- \(\displaystyle -14\)
What about the product of two negative numbers?
Product of Negative Numbers.
The product of two negative numbers is positive.
This rule is the hardest one to understand. Try reading the Note below to see if it convinces you.
Note 7.3.4.
Earlier we argued that erasing a debt of $500 had the same effect as adding $500 to your net worth. This example helped us understand that subtracting a negative number is the same as adding a positive number.
Now imagine that three debts of $500 are erased from your balance sheet. This is equivalent to increasing your net worth by $1500. We express this mathematically as
\begin{equation*}
-3(-500) = 1500
\end{equation*}
Example 7.3.5.
- \(\displaystyle (-4)(-5) = 20\)
- \(\displaystyle -1.5(-8) = 12\)
Checkpoint 7.3.6.
Multiply.
- \(\displaystyle (-9)(-7)\)
- \(\displaystyle (-6)(-0.4)\)
Answer.
- \(\displaystyle 63\)
- \(\displaystyle 2.4\)
We already know that the product of two positive numbers is positive. And here is another reasonable fact:
Products with Zero.
Zero times any number, positive or negative, is zero.
For example,
\begin{equation*}
0(-28) = 0 ~~~~ \text{and} ~~~~ -17(0) = 0
\end{equation*}
Combining all these facts, we get the following set of rules.
Products of Signed Numbers.
- The product of two numbers with the same sign is positive.
- The product of two numbers with opposite signs is negative.
- The product of any number and zero is zero.
Subsection 7.3.2 Division
Division is the opposite operation for multiplication. Every division fact can be rewritten as an equivalent multiplication fact. For example,
\begin{equation*}
\dfrac{12}{4} = \alert{3} ~~~~ \text{because} ~~~~ 4 \cdot \alert{3} = 12
\end{equation*}
We’ll use this idea to investigate division of signed numbers.
Activity 7.3.1. Quotients.
Each division problem below is rewritten as a multiplication problem.
- \(\dfrac{-20}{5} = \fillinmath{XXX}~~\) because \(~~5 \cdot \fillinmath{XXX} = -20\)
- \(\dfrac{-14}{-2} = \fillinmath{XXX}~~\) because \(~~-2 \cdot \fillinmath{XXX} = -14\)
-
In the first multiplication problem, should the blank be replaced by a positive number or a negative number?Why?
- Fill in the blanks:\begin{equation*} 5 \cdot \fillinmath{XXX} = -20, ~~~~ \text{so} ~~~~ \dfrac{-20}{5} = \fillinmath{XXX} \end{equation*}
-
From this example, we see that:The quotient of a negative number and a positive number is .
-
In the second multiplication problem, should the blank be replaced by a positive number or a negative number?Why?
- Fill in the blanks:\begin{equation*} -2 \cdot \fillinmath{XXX} = -14, ~~~~ \text{so} ~~~~ \dfrac{-14}{-2} = \fillinmath{XXX} \end{equation*}
-
From this example, we see that:The quotient of two negative numbers is .
- Compute the quotients.
- \(\displaystyle \dfrac{27}{-3}\)
- \(\displaystyle \dfrac{-28}{-7}\)
- \(\displaystyle -56 \div (-7)\)
- \(\displaystyle -12 \div 2\)
What about quotients involving zero? We know that zero times any number is zero, so what does that mean for division?
Activity 7.3.2. Quotients Involving Zero.
- We’ll first consider two examples where zero is divided by a number. Rewrite each quotient as a product.\begin{equation*} \dfrac{0}{8} = \blert{?} ~~~~ \text{is equivalent to:} ~~ \fillinmath{XXX} \end{equation*}\begin{equation*} \dfrac{0}{-5} = \blert{?} ~~~~ \text{is equivalent to:} ~~ \fillinmath{XXX} \end{equation*}
- What number can we use to replace the question marks and make true statements? There is only one possible answer.\begin{equation*} \dfrac{0}{8} = \blert{?} ~~~~ \text{because} ~~~~ 8 \cdot \blert{?} = 0 \end{equation*}and\begin{equation*} \dfrac{0}{8} = \blert{?} ~~~~ \text{because} ~~~~ 8 \cdot \blert{?} = 0 \end{equation*}
- From these examples, we see that:The quotient of zero divided by any number (except zero) is .
- Next we’ll look at two examples in which a number is divided by zero. Again, begin by rewriting each quotient as a product.\begin{equation*} \dfrac{15}{0} = \blert{?} ~~~~ \text{is equivalent to:} ~~ \fillinmath{XXX} \end{equation*}\begin{equation*} \dfrac{-4}{0} = \blert{?} ~~~~ \text{is equivalent to:} ~~ \fillinmath{XXX} \end{equation*}
- What number can we use to replace the question marks and make true statements?
-
Because zero times any number is zero, we cannot find any number that will result in a product of 15 or a product of 4. There is no solution to either multiplication problem, and consequently there are no solutions to the division problems either. We say that division by zero is undefined.The quotient of any number divided by zero is .
- Compute the quotients.
- \(\displaystyle \dfrac{0}{-5}\)
- \(\displaystyle \dfrac{-18}{0}\)
- \(\displaystyle \dfrac{-83}{0}\)
- \(\displaystyle \dfrac{0}{-83}\)
We summarize the rules for division as follows.
Rules for Dividing Signed Numbers.
- The quotient of two numbers with the same sign is positive.
- The quotient of two numbers with opposite signs is negative.
- Zero divided by any number (except zero) is zero.
- The quotient of any number divided by zero is undefined.
Subsection 7.3.3 Powers of Negative Numbers
Recall that an exponent indicates repeated multiplication. For example,
\begin{equation*}
4^3 ~~~~ \text{means} ~~~~ 4 \cdot 4 \cdot 4
\end{equation*}
where 4 is called the base and 3 is called the exponent. We read \(4^3\) as "4 to the third power" or "4 cubed." We can also compute powers of negative numbers.
Powers of Negative NUmbers.
To show that a negative number is raised to a power, we must enclose the negative number in parentheses.
For example, to indicate the square of 5, we write
\begin{equation*}
(-5)^2 = (-5)(-5) = 25 ~~~~~~ \blert{\text{Exponent applies to}~ (-5).}
\end{equation*}
If the negative number is not enclosed in parentheses, then the exponent applies only to the positive number. The negative sign tells us that the power is negative. For example,
\begin{equation*}
-5^2 = -(5 \cdot 5) = -25 ~~~~~~ \blert{\text{Exponent applies only to}~ 5.}
\end{equation*}
Look carefully at how the placement of parentheses changes the meaning of the expressions in the next Example.
Example 7.3.7.
Compute each power.
- \(\displaystyle -4^2\)
- \(\displaystyle (-4)^2\)
- \(\displaystyle -(4)^2\)
- \(\displaystyle (-4^2)\)
Solution.
- Only 4 is squared:\begin{equation*} -4^2 = -4 \cdot 4 = -16 \end{equation*}
- The negative number is squared:\begin{equation*} (-4)^2 = (-4)(-4) = 16 \end{equation*}
- Only 4 is squared:\begin{equation*} -(4)^2 = -(4)(4) = -16 \end{equation*}
- Only 4 is squared, and the entire expression appears within parentheses:\begin{equation*} (-4^2) = (-4 \cdot 4) = -16 \end{equation*}
Checkpoint 7.3.8.
Write each power as a repeated product, then compute.
- \(\displaystyle (-2)^4\)
- \(\displaystyle -2^4\)
- \(\displaystyle (-2)^3\)
Answer.
- \(\displaystyle 16\)
- \(\displaystyle -16\)
- \(\displaystyle -8\)
Subsection 7.3.4 Vocabulary
- product
- quotient
- power
- expponent
Exercises 7.3.5 Practice 7-3
Exercise Group.
For Problems 1-8, multiply or divide, if possible.
1.
- \(\displaystyle 3(-6)\)
- \(\displaystyle -4(-8)\)
2.
- \(\displaystyle -9(-2)\)
- \(\displaystyle -6 \cdot 8\)
3.
- \(\displaystyle -7 \cdot 4\)
- \(\displaystyle 5(-9)\)
4.
- \(\displaystyle -36 \div 4\)
- \(\displaystyle -42 \div (-6)\)
5.
- \(\displaystyle -56 \div (-8)\)
- \(\displaystyle 70 \div (-10)\)
6.
- \(\displaystyle 24 \div (-8)\)
- \(\displaystyle -63 \div 7\)
7.
- \(\displaystyle 0 \div (-12)\)
- \(\displaystyle 0(-12)\)
8.
- \(\displaystyle -12 \div 0\)
- \(\displaystyle -12 \cdot 0\)
9.
- \(\displaystyle 12 \div (-12)\)
- \(\displaystyle 12(-12)\)
10.
- \(\displaystyle \dfrac{-84}{-6}\)
- \(\displaystyle \dfrac{-110}{5}\)
11.
- \(\displaystyle \dfrac{96}{-6}\)
- \(\displaystyle \dfrac{-95}{-5}\)
12.
- \(\displaystyle \dfrac{0}{-7}\)
- \(\displaystyle \dfrac{-7}{0}\)
13.
Use the idea of repeated addition to explain why the product of a positive number and a negative number is negative.
14.
Use ideas about money to explain why the product of two negative numbers is positive.
15.
Delbert says that "two negatives make a positive." For which operations is he correct: addition, subtraction, multiplication, division.
16.
True or False:
- The product of any number and zero is undefined.
- The quotient of any number divided by zero is undefined.
- The quotient of zero divided by any number is undefined.
- The sum of any number and its opposite is undefined.
17.
Fill in each blank with "is always positive," "is always negative," or "could be either positive or negative."
- The sum of two negative numbers .
- The difference of two negative numbers .
- The product of two negative numbers .
- The quotient of two negative numbers .
18.
Fill in each blank with "is always positive," "is always negative," or "could be either positive or negative."
- The sum of two a positive number and a negative number .
- The difference of a positive number and a negative number .
- The product of a positive number and a negative number .
- The quotient of a positive number and a negative number .
Exercise Group.
For Problems 19-24,
- Write a product or quotient of signed numbers to describe the story.
- Evaluate the product or quotient to answer the question.
19.
Whitney is climbing down a sheer cliff. She has spaced pitons vertically every 6 meters. What is her net elevation change after she has descended to the eighth piton from the top?
20.
The bank erroneously charged Utako with five withdrawals of $400 each. If her balance is actually $0, what does the bank believe her balance is?
21.
The temperature on the ice planet Hoth dropped 115\(\deg\) in just 4 hours. What was the average temperature change per hour?
22.
Bertha lost 30 pounds in 8 weeks. What was Bertha’s average change in weight per week?
23.
The U.S. balance book shows an entry of $100,000,000,000 representing the failure of numerous savings and loan institutions. If this amount were to be divided equally among the 100 million adult citizens, how much of the balance would be assigned to each?
24.
Petra implements a scheme that successfully cuts her firm’s $250,000 debt in half. What is the resulting balance?
Exercise Group.
For Problems 25-34, which operation is indicated in each expression? Simplify the expression.
25.
- \(\displaystyle -6(-8)\)
- \(\displaystyle -6 - 8\)
26.
- \(\displaystyle 5 -(-9)\)
- \(\displaystyle -5 (-9)\)
27.
- \(\displaystyle 12 - (-4)\)
- \(\displaystyle 12 \div (-4)\)
28.
- \(\displaystyle -36 \div 9\)
- \(\displaystyle -36 - 9\)
29.
- \(\displaystyle -40 - (-8)\)
- \(\displaystyle -40 \div (-8)\)
30.
- \(\displaystyle 24 + (-6)\)
- \(\displaystyle 24 \div (-6)\)
31.
- \(\displaystyle -3 + (-8)\)
- \(\displaystyle -3(-8)\)
32.
- \(\displaystyle -2 (-13)\)
- \(\displaystyle -2 - (-13)\)
33.
- \(\displaystyle -9(7)\)
- \(\displaystyle -9 + 7\)
34.
- \(\displaystyle 8-12\)
- \(\displaystyle 8(-12)\)
Exercise Group.
For Problems 35-40, write each division fact as an equivalent multiplication fact.
35.
\(-40 \div 8 = -5\)36.
\(-80 \div (-16) = 5\)37.
\(\dfrac{56}{-4} = -14\)38.
\(\dfrac{-72}{-3} = 24\)39.
\(\dfrac{-196}{-7} = 28\)40.
\(\dfrac{-72}{12} = -6\)Exercise Group.
For Problems 41-48, use your calculator to multiply or divide. Round your answers to two decimal places if necessary.
41.
\(-18 (-23)\)42.
\(374 \div (-22)\)43.
\(-434 \div (-14)\)44.
\(25(-19)\)45.
\(\dfrac{288}{-16}\)46.
\(\dfrac{-312}{-26}\)47.
\((-0.3)(26.1)\)48.
\((2.8)(-5.4)\)Exercise Group.
For Problems 49-52, compute each power.
49.
- \(\displaystyle (-3)^2\)
- \(\displaystyle (-2)^3\)
- \(\displaystyle -3^2\)
- \(\displaystyle -2^3\)
50.
- \(\displaystyle -4^2\)
- \(\displaystyle -4^3\)
- \(\displaystyle (-4)^2\)
- \(\displaystyle (-4)^3\)
51.
- \(\displaystyle (-1)^3\)
- \(\displaystyle (-1)^4\)
- \(\displaystyle -1^4\)
- \(\displaystyle -1^5\)
52.
- \(\displaystyle (-5)^3\)
- \(\displaystyle (-9)^2\)
- \(\displaystyle -8^2\)
- \(\displaystyle -3^3\)
53.
- Compute each power.
- \(\displaystyle (-2)^1\)
- \(\displaystyle (-2)^2\)
- \(\displaystyle (-2)^3\)
- \(\displaystyle (-2)^4\)
- \(\displaystyle (-2)^5\)
- \(\displaystyle (-2)^6\)
- Do you see a pattern developing? What is the connection between the exponent and the sign of the answer? The sign of the power is if the exponent is .
54.
- Explain why \((-3)^4\) and \(-3^4\) are different numbers.
- Explain why \((-3)^3\) and \(-3^3\) are the same number.
