3.4 Multiply and divide integers (Page 2/3)

Prealgebra Page 2 / 3

\begin{matrix} 5 \cdot 3 = 15 so 15 \div 3 = 5 & −5 (3) = −15 so −15 \div 3 = −5 \\ (−5) (−3) = 15 so 15 \div (−3) = −5 & 5 (−3) = −15 so −15 \div −3 = 5 \end{matrix}

Division of signed numbers follows the same rules as multiplication. When the signs are the same, the quotient is positive, and when the signs are different, the quotient is negative.

Division of signed numbers

The sign of the quotient of two numbers depends on their signs.

Same signs	Quotient
•Two positives •Two negatives	Positive Positive

Different signs	Quotient
•Positive&negative •Negative&positive	Negative Negative

Remember, you can always check the answer to a division problem by multiplying.

Divide each of the following:

ⓐ $−27 \div 3$
ⓑ $−100 \div (−4)$

Solution

ⓐ
	$–27 \div 3$
Divide, noting that the signs are different and so the product is negative.	$–9$

ⓑ
	$–100 \div (–4)$
Divide, noting that the signs are the same and so the product is positive.	$25$

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Divide:

ⓐ $−42 \div 6$
ⓑ $−117 \div (−3)$

ⓐ − 7
ⓑ 39

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Divide:

ⓐ $−63 \div 7$
ⓑ $−115 \div (−5)$

ⓐ − 9
ⓑ 23

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Just as we saw with multiplication, when we divide a number by $1,$ the result is the same number. What happens when we divide a number by $−1 ?$ Let’s divide a positive number and then a negative number by $−1$ to see what we get.

\begin{matrix} 8 \div (−1) & −9 \div (−1) \\ −8 & 9 \\ −8 is the opposite of 8 & 9 is the opposite of −9 \end{matrix}

When we divide a number by, $−1$ we get its opposite.

Division by $−1$

Dividing a number by $−1$ gives its opposite.

a \div (−1) = −a

Divide each of the following:

ⓐ $16 \div (−1)$
ⓑ $−20 \div (−1)$

Solution

ⓐ
	$16 \div (–1)$
The dividend, 16, is being divided by –1.	$–16$
Dividing a number by –1 gives its opposite.
Notice that the signs were different, so the result was negative.

ⓑ
	$–20 \div (–1)$
The dividend, –20, is being divided by –1.	$20$
Dividing a number by –1 gives its opposite.

Notice that the signs were the same, so the quotient was positive.

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Divide:

ⓐ $6 \div (−1)$
ⓑ $−36 \div (−1)$

ⓐ − 6
ⓑ 36

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Divide:

ⓐ $28 \div (−1)$
ⓑ $−52 \div (−1)$

ⓐ − 28
ⓑ 52

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Simplify expressions with integers

Now we’ll simplify expressions that use all four operations–addition, subtraction, multiplication, and division–with integers. Remember to follow the order of operations.

$Simplify: 7 (−2) + 4 (−7) - 6 .$

Solution

We use the order of operations. Multiply first and then add and subtract from left to right.

	$7 (−2) + 4 (−7) −6$
Multiply first.	$−14 + (−28) −6$
Add.	$−42 - 6$
Subtract.	$−48$

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Simplify:

$8 (−3) + 5 (−7) −4$

− 63

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Simplify:

$9 (−3) + 7 (−8) - 1$

− 84

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Simplify:

ⓐ ${(−2)}^{4}$
ⓑ ${−2}^{4}$

Solution

The exponent tells how many times to multiply the base.

ⓐ The exponent is $4$ and the base is $−2 .$ We raise $−2$ to the fourth power.

	${(−2)}^{4}$
Write in expanded form.	$(−2) (−2) (−2) (−2)$
Multiply.	$4 (−2) (−2)$
Multiply.	$- 8 (−2)$
Multiply.	$16$

ⓑ The exponent is $4$ and the base is $2 .$ We raise $2$ to the fourth power and then take the opposite.

	$- 2^{4}$
Write in expanded form.	$- (2 \cdot 2 \cdot 2 \cdot 2)$
Multiply.	$- (4 \cdot 2 \cdot 2)$
Multiply.	$- (8 \cdot 2)$
Multiply.	$- 16$

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Simplify:

ⓐ ${(−3)}^{4}$
ⓑ ${−3}^{4}$

ⓐ 81
ⓑ −81

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Simplify:

ⓐ ${(−7)}^{2}$
ⓑ $- 7^{2}$

ⓐ 49
ⓑ −49

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$Simplify: 12 - 3 (9 - 12) .$

Solution

According to the order of operations, we simplify inside parentheses first. Then we will multiply and finally we will subtract.

	$12 - 3 (9 - 12)$
Subtract the parentheses first.	$12 - 3 (−3)$
Multiply.	$12 - (−9)$
Subtract.	$21$

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Simplify:

$17 - 4 (8 - 11)$

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Simplify:

$16 - 6 (7 - 13)$

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Simplify: $8 (−9) \div {(−2)}^{3} .$

Solution

We simplify the exponent first, then multiply and divide.

	$8 (−9) \div {(−2)}^{3}$
Simplify the exponent.	$8 (−9) \div (−8)$
Multiply.	$−72 \div (−8)$
Divide.	$9$

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Simplify:

$12 (−9) \div {(−3)}^{3}$

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Source: OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9

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3.4 Multiply and divide integers (Page 2/3)

Division of signed numbers

Solution

Division by −1

Solution

Simplify expressions with integers

Solution

Solution

Solution

Solution

Questions & Answers

Read also:

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Division by $−1$