Multiplying and Dividing Integers

The rules of signs for multiplying and dividing integers, squares, cubes and roots of negative numbers, and the four operations together in the correct order.

Before you start

The integers are $\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$ Adding a negative integer is a move to the left on the number line: $(-4) + (-4) = -8$ .

Multiplying whole numbers is repeated addition: $3 \times 4 = 4 + 4 + 4 = 12$ .

Multiplying Integers

$3 \times (-4)$ means three lots of $-4$ added together: $3 \times (-4) = (-4) + (-4) + (-4) = -12$ . On the number line, this is three jumps of 4 to the left, starting from 0.

Two numbers can be multiplied in either order, so $(-4) \times 3 = 3 \times (-4) = -12$ too. A positive integer and a negative integer have a negative product.

What about a negative integer times a negative integer? Read down this table. Each time the first number goes down by 1, the product goes up by 4.

Multiplication	Product
$3 \times (-4)$	$-12$
$2 \times (-4)$	$-8$
$1 \times (-4)$	$-4$
$0 \times (-4)$	$0$
$(-1) \times (-4)$	$4$
$(-2) \times (-4)$	$8$
$(-3) \times (-4)$	$12$

For the pattern to carry on after $0 \times (-4) = 0$ , the last three products must be 4, 8 and 12. Two negative integers have a positive product.

Rule of signs for multiplication

To multiply two integers, multiply their numerical values (the numbers without their signs). Then give the product its sign from the table.

×	positive	negative
positive	positive	negative
negative	negative	positive

The sign of the first number is at the side and the sign of the second number is along the top.

Same signs give a positive product. Different signs give a negative product. The product of any integer and 0 is 0.

Example 1: Multiplying two integers

Evaluate each of the following.
(a) $6 \times (-5)$
(b) $(-6) \times (-7)$
(c) $(-9) \times 4$
(d) $(-61) \times 0$

Solution

(a) $6 \times 5 = 30$ . The signs are different, so the product is negative: $6 \times (-5) = -30$ .

(b) $6 \times 7 = 42$ . The signs are the same, so the product is positive: $(-6) \times (-7) = 42$ .

(d) Any integer multiplied by 0 gives 0: $(-61) \times 0 = 0$ .

The values are (a) $-30$ , (b) $42$ , (c) $-36$ and (d) $0$ .

To multiply three or more integers, multiply them two at a time. Each pair of negative factors gives a positive product, so the sign can be found by counting the negative factors.

Sign of a product of several integers

When no factor is 0, the product is positive if the number of negative factors is even, and negative if the number of negative factors is odd.

Example 2: Three or more integers

Evaluate each of the following.
(a) $(-4) \times (-12) \times (-2)$
(b) $(-1) \times (-2) \times 3 \times (-5) \times (-4)$

Solution

(a) Two at a time: $(-4) \times (-12) = 48$ , and $48 \times (-2) = -96$ .

Check by counting: there are three negative factors, an odd number, so the product is negative, and $4 \times 12 \times 2 = 96$ .

(b) There are four negative factors, an even number, so the product is positive. $1 \times 2 \times 3 \times 5 \times 4 = 120$ .

The values are (a) $-96$ and (b) $120$ .

Squares, Cubes and Roots of Negative Numbers

A square has two equal factors and a cube has three: $(-3)^{2} = (-3) \times (-3) = 9$ and $(-2)^{3} = (-2) \times (-2) \times (-2) = -8$ .

Rule

The square of a negative number is positive. The cube of a negative number is negative.

Take care with brackets

$(-3)^{2}$ and $-3^{2}$ are different. In $(-3)^{2}$ the brackets show that $-3$ is squared: $(-3)^{2} = 9$ . In $-3^{2}$ only the 3 is squared, and the minus sign is put in front afterwards: $-3^{2} = -(3 \times 3) = -9$ .

Square roots. $4^{2} = 16$ and $(-4)^{2} = 16$ , so 16 has two square roots, 4 and $-4$ . The sign $\sqrt{\phantom{9}}$ stands for the positive one only: $\sqrt{16} = 4$ . The negative square root is written $-\sqrt{16} = -4$ . 0 has only one square root, 0, so $\sqrt{0} = 0$ . A negative number has no square root, because the square of a number is never negative.

Cube roots. $(-2)^{3} = -8$ , so $\sqrt[3]{-8} = -2$ . Every number has exactly one cube root, and the cube root of a negative number is negative.

Example 3: Powers and roots

Evaluate each of the following.
(a) $(-5)^{3}$
(b) $-7^{2}$
(c) $(-3)^{2} \times (-8)$
(d) $-\sqrt{49} \times \sqrt[3]{-27}$

Solution

(a) $(-5)^{3} = (-5) \times (-5) \times (-5)$ . Three negative factors give a negative product, and $5 \times 5 \times 5 = 125$ , so $(-5)^{3} = -125$ .

(b) There are no brackets, so only the 7 is squared: $-7^{2} = -(7 \times 7) = -49$ .

(d) $\sqrt{49} = 7$ , so $-\sqrt{49} = -7$ . Since $(-3)^{3} = -27$ , $\sqrt[3]{-27} = -3$ . Then $(-7) \times (-3) = 21$ .

The values are (a) $-125$ , (b) $-49$ , (c) $-72$ and (d) $21$ .

Dividing Integers

Division is the reverse of multiplication. Since $3 \times (-4) = -12$ , we have $(-12) \div 3 = -4$ and $(-12) \div (-4) = 3$ . Since $(-3) \times (-4) = 12$ , we have $12 \div (-4) = -3$ . So the signs follow the same rule as in multiplication.

Rule of signs for division

To divide an integer by an integer other than 0, divide their numerical values. Same signs give a positive quotient. Different signs give a negative quotient, as in the table of signs for multiplication.

Division and zero

0 divided by any integer other than 0 is 0. For example, $0 \div (-8) = 0$ , because $0 \times (-8) = 0$ .

Division by 0 is not defined. $(-55) \div 0$ would be a number that gives $-55$ when it is multiplied by 0, but every number multiplied by 0 gives 0. $0 \div 0$ is not defined either: every number gives 0 when it is multiplied by 0, so there is no one answer.

Example 4: Dividing integers

Evaluate each of the following.
(a) $33 \div (-3)$
(b) $(-65) \div (-5)$
(c) $(-144) \div 24$
(d) $56 \div (-\sqrt{64})$

Solution

(a) $33 \div 3 = 11$ . The signs are different, so the quotient is negative: $33 \div (-3) = -11$ . Check: $(-11) \times (-3) = 33$ .

(b) $65 \div 5 = 13$ . The signs are the same, so the quotient is positive: $(-65) \div (-5) = 13$ . Check: $13 \times (-5) = -65$ .

(d) $\sqrt{64} = 8$ , so $56 \div (-\sqrt{64}) = 56 \div (-8) = -7$ .

The values are (a) $-11$ , (b) $13$ , (c) $-6$ and (d) $-7$ .

The Four Operations Together

When a calculation has more than one operation, integers follow the same order of operations as whole numbers.

Order of operations

Work out what is inside brackets, starting with the innermost brackets.
Work out powers and roots.
Multiply and divide, from left to right.
Add and subtract, from left to right.

Brackets round a single negative number, as in $(-3)$ , only hold its sign. There is nothing to work out inside them.

Example 5: Multiplication and division before addition and subtraction

Evaluate each of the following.
(a) $(-10) \times 5 \div 2$
(b) $51 \div (-3) + 12 \times (-3)$
(c) $1 - (-75) \div 5 - (-4)$

Solution

(a) Multiply and divide from left to right: $(-10) \times 5 \div 2 = (-50) \div 2 = -25$ .

(b) Divide and multiply first: $51 \div (-3) = -17$ and $12 \times (-3) = -36$ . Then add: $(-17) + (-36) = -53$ .

(c) Divide first: $(-75) \div 5 = -15$ , which leaves $1 - (-15) - (-4)$ . Subtracting a number is the same as adding its opposite, so subtracting a negative number is the same as adding the positive number: $1 - (-15) - (-4) = 1 + 15 + 4 = 20$ .

The values are (a) $-25$ , (b) $-53$ and (c) $20$ .

Example 6: Brackets, powers and roots

Evaluate each of the following.
(a) $11 \times (-3)^{2} - 22$
(b) $(\sqrt{81} + 3^{2}) \times 21 \div (-7)$
(c) $20 - [(-6) + (4 - 9) \times 2] \div (-4)$

Solution

(a) Power first: $(-3)^{2} = 9$ . Then multiply: $11 \times 9 = 99$ . Then subtract: $99 - 22 = 77$ .

(b) Brackets first: $\sqrt{81} + 3^{2} = 9 + 9 = 18$ . Then from left to right: $18 \times 21 = 378$ , and $378 \div (-7) = -54$ .

(c) Innermost brackets first: $4 - 9 = -5$ . Inside the square brackets, multiply before adding: $(-5) \times 2 = -10$ , and $(-6) + (-10) = -16$ . Then divide: $(-16) \div (-4) = 4$ . Then subtract: $20 - 4 = 16$ .

The values are (a) $77$ , (b) $-54$ and (c) $16$ .

Word Problems

A fall, a loss, a withdrawal or a penalty is a negative change. The same change made several times is a multiplication.

Example 7: A temperature that falls steadily

A cold room is at 4 °C when its cooling is switched on. Its temperature then falls by 3 °C every hour.
(a) Find the temperature after 6 hours.
(b) After how many hours is the temperature $-20$ °C?

Solution

A fall of 3 °C is a change of $-3$ °C each hour.

(a) Change in 6 hours: $6 \times (-3) = -18$ . Temperature: $4 + (-18) = -14$ .

(b) Change needed: $(-20) - 4 = -24$ . Number of hours: $(-24) \div (-3) = 8$ .

(a) The temperature after 6 hours is $-14$ °C. (b) The temperature is $-20$ °C after 8 hours.

Example 8: A score with penalties

In a quiz of 20 questions, each correct answer scores 4 marks, each wrong answer scores $-2$ marks and each question left out scores 0 marks. Ravi has 5 correct answers and 12 wrong answers, and leaves out 3 questions. Find his score.

Solution

Score $= 5 \times 4 + 12 \times (-2) + 3 \times 0$

Multiply first: $20 + (-24) + 0$

Then add from left to right: $20 + (-24) = -4$ , and $(-4) + 0 = -4$ .

Ravi scores $-4$ marks.

Common Mistakes

Using the rule of signs when adding. The rule of signs is for multiplying and dividing: $(-3) \times (-5) = 15$ , but $(-3) + (-5) = -8$ .
Leaving out the brackets. $(-4)^{2} = 16$ , but $-4^{2} = -16$ .
Giving two values for $\sqrt{16}$ . 16 has two square roots, but $\sqrt{16}$ means the positive one, 4.
Working from left to right whatever the operations. $2 - 3 \times (-4)$ is $2 - (-12) = 14$ . It is not $(-1) \times (-4) = 4$ .
Writing $(-5) \div 0 = 0$ . It is $0 \div (-5)$ that is 0. $(-5) \div 0$ is not defined.

Practice

Work each question on paper before you open its solution.

Question 1

Write $5 \times (-7)$ as a repeated addition and find its value.

ShowHide solution

$5 \times (-7)$ means five lots of $-7$ added together.

$5 \times (-7) = (-7) + (-7) + (-7) + (-7) + (-7) = -35$

$5 \times (-7) = -35$

Question 2

Evaluate each of the following.
(a) $8 \times (-5)$
(b) $(-12) \times (-11)$
(c) $(-45) \div 5$
(d) $(-36) \div (-9)$

ShowHide solution

(a) $8 \times 5 = 40$ , and the signs are different: $8 \times (-5) = -40$ .

(b) $12 \times 11 = 132$ , and the signs are the same: $(-12) \times (-11) = 132$ .

(d) $36 \div 9 = 4$ , and the signs are the same: $(-36) \div (-9) = 4$ .

The values are (a) $-40$ , (b) $132$ , (c) $-9$ and (d) $4$ .

Question 3

Evaluate each of the following.
(a) $(-2) \times (-3) \times (-4)$
(b) $(-5) \times 2 \times (-1) \times (-3) \times (-2)$
(c) $(-7) \times 13 \times 0 \times (-2)$

ShowHide solution

(a) There are three negative factors, an odd number, so the product is negative. $2 \times 3 \times 4 = 24$ .

(b) There are four negative factors, an even number, so the product is positive. $5 \times 2 \times 1 \times 3 \times 2 = 60$ .

The values are (a) $-24$ , (b) $60$ and (c) $0$ .

Question 4

Evaluate each of the following.
(a) $(-6)^{2}$
(b) $-6^{2}$
(c) $(-4)^{3}$
(d) $\sqrt[3]{-125}$
(e) $-\sqrt{81}$

ShowHide solution

(a) $(-6)^{2} = (-6) \times (-6) = 36$

(b) Only the 6 is squared: $-6^{2} = -(6 \times 6) = -36$

(d) $(-5)^{3} = -125$ , so $\sqrt[3]{-125} = -5$ .

(e) $\sqrt{81} = 9$ , so $-\sqrt{81} = -9$ .

The values are (a) $36$ , (b) $-36$ , (c) $-64$ , (d) $-5$ and (e) $-9$ .

Question 5

(a) Write down the two square roots of 49.
(b) Write down the value of $\sqrt{49}$ .
(c) Explain why $-49$ has no square root.

ShowHide solution

(a) $7^{2} = 49$ and $(-7)^{2} = (-7) \times (-7) = 49$ , so 49 has two square roots.

(b) The sign $\sqrt{\phantom{9}}$ stands for the positive square root only.

(c) The square of a positive number is positive, the square of a negative number is positive, and $0^{2} = 0$ . So the square of a number is never negative.

(a) 7 and $-7$ (b) $\sqrt{49} = 7$ (c) No number has a square of $-49$ , because the square of a number is never negative.

Question 6

Evaluate each of the following where possible.
(a) $0 \div (-12)$
(b) $(-12) \div 0$
(c) $(-96) \div (-8) \div (-4)$

ShowHide solution

(a) 0 divided by an integer other than 0 is 0.

(b) Division by 0 is not defined, so this has no value.

(a) $0$ (b) Not defined (c) $-3$

Question 7

Evaluate each of the following.
(a) $12 \div (-3) - 2 \times 5$
(b) $(-2) \times (3 - 7) + 10 \div (-5)$

ShowHide solution

(a) Divide and multiply first: $12 \div (-3) = -4$ and $2 \times 5 = 10$ . Then subtract: $(-4) - 10 = -14$ .

(b) Brackets first: $3 - 7 = -4$ . Then multiply and divide: $(-2) \times (-4) = 8$ and $10 \div (-5) = -2$ . Then add: $8 + (-2) = 6$ .

The values are (a) $-14$ and (b) $6$ .

Question 8

Evaluate each of the following.
(a) $350 \div (-5)^{2} - (-6)^{3}$
(b) $\sqrt[3]{-64} \times \sqrt{25} - (-2)^{3}$
(c) $[(-15) - (-3) + (-15)^{2}] \div 3 \times (-4)$
(d) $(-5) \times [7 - (3 - 9)^{2} \div 4] - 12$

ShowHide solution

(a) Powers first: $(-5)^{2} = 25$ and $(-6)^{3} = -216$ . Then divide: $350 \div 25 = 14$ . Then subtract: $14 - (-216) = 14 + 216 = 230$ .

(b) Roots and powers first: $\sqrt[3]{-64} = -4$ , $\sqrt{25} = 5$ and $(-2)^{3} = -8$ . Then multiply: $(-4) \times 5 = -20$ . Then subtract: $(-20) - (-8) = (-20) + 8 = -12$ .

(c) Inside the brackets, the power first: $(-15)^{2} = 225$ . Then from left to right: $(-15) - (-3) = -12$ , and $(-12) + 225 = 213$ . Outside the brackets, from left to right: $213 \div 3 = 71$ , and $71 \times (-4) = -284$ .

(d) Innermost brackets first: $3 - 9 = -6$ . Inside the square brackets, the power first: $(-6)^{2} = 36$ . Then divide: $36 \div 4 = 9$ . Then subtract: $7 - 9 = -2$ . Outside the brackets, multiply: $(-5) \times (-2) = 10$ . Then subtract: $10 - 12 = -2$ .

The values are (a) $230$ , (b) $-12$ , (c) $-284$ and (d) $-2$ .

Question 9

Mr Lim withdraws the same amount of money from his bank account every week, and nothing else goes into or out of the account. After 6 weeks, the balance is $450 less than it was at the start.
(a) Write the change in the balance over the 6 weeks as an integer.
(b) Find the change in the balance each week. How much does he withdraw each week?

ShowHide solution

(a) The balance has gone down by 450 dollars, so the change is $-450$ .

(b) The change is the same every week, so divide by 6: $(-450) \div 6 = -75$ .

(a) The change in the balance is $-450$ dollars. (b) The balance changes by $-75$ dollars each week, so he withdraws $75 each week.

Question 10

A drinks stall makes a loss of $45 on each rainy day and a profit of $60 on each sunny day. One week it is open for 6 days, of which 4 are rainy and 2 are sunny.
(a) Write the profit on a rainy day as an integer.
(b) Find the total profit for the 6 days.
(c) Find the average profit per day.

ShowHide solution

(a) A loss is a profit below zero, so the profit on a rainy day is $-45$ dollars.

(b) Total profit $= 4 \times (-45) + 2 \times 60$ . Multiply first: $(-180) + 120 = -60$ .

(a) $-45$ dollars (b) The total profit is $-60$ dollars, which is a loss of $60. (c) The average profit is $-10$ dollars per day, which is a loss of $10 a day.

Question 11

In a game, a player gains 5 points for every target hit and loses 3 points for every miss. Mei Ling hits 7 targets and her final score is $-1$ . How many times does she miss?

ShowHide solution

Points from the hits: $7 \times 5 = 35$

Points from the misses: $(-1) - 35 = -36$

Each miss scores $-3$ points, so the number of misses is $(-36) \div (-3) = 12$ .

Check: $7 \times 5 + 12 \times (-3) = 35 + (-36) = -1$

Mei Ling misses 12 times.

Key Points

Multiplying or dividing two integers: same signs give a positive answer, different signs give a negative answer.
A product of integers other than 0 is positive when the number of negative factors is even, and negative when it is odd.
$(-3)^{2} = 9$ but $-3^{2} = -9$ . The cube of a negative number is negative: $(-2)^{3} = -8$ .
16 has two square roots, 4 and $-4$ , and $\sqrt{16}$ means 4 only. A negative number has no square root, and its cube root is negative: $\sqrt[3]{-8} = -2$ .
Any integer multiplied by 0 gives 0, and 0 divided by an integer other than 0 is 0. Division by 0 is not defined.
Order of operations: brackets; then powers and roots; then multiplication and division from left to right; then addition and subtraction from left to right.