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Integers

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Integer Unit Topics· Integer Definition · Absolute Value · Comparing and Ordering Integers· Integer Addition· Turning Subtraction Into Addition· Adding and Subtracting Integers Review· Multiplying Integers· Dividing Integers· Powers of Integers· Rules for Exponents

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Integer Definition

Return toTable ofContents

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1 Do you know what an integer is?

Yes

No

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{...-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7...}

Definition of Integer:

The set of natural numbers, their opposites and zero.

Define Integer

Examples of Integer:

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integer not an integer

Classify each number as an integer, or not.5

-6

0 -21 -65 1 3.2-6.32

92.34437 x 103

½ ¾3¾ π 5

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-1 0-2-3-4-5 1 2 3 4 5

Integers on the number line

NegativeIntegers

PositiveIntegers

Numbers to the left of zero are less than zero

Numbers to the right of zero are greater than zero

Zero is neitherpositive or negative

`

Zero

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2 Which of the following are examples of integers?

A 0B -8C -4.5D 7E 1/2

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3 Which of the following are examples of integers?

A 1/2B 6C -4D 0.75E 25%

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You might hear "And the quarterback is sacked for a loss of 7 yards."

This can be represented as an integer: -7

Or, "The total snow fall this year has been 9 inches more than normal."

This can be represented as in integer: +9 or 9

Integers can represent everyday situations

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Integers In Our World

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1. Spending $6

2. Gain of 11 pounds

3. Depositing $700

4. 10 degrees below zero

5. 8 strokes under par (par = 0)

6. 350 feet above sea level

Write an integer to represent each situation:

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4 Which of the following integers best represents the following scenario:

The effect on your wallet when you spend 10 dollars.

A -10B 10C 0D +/- 10

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Earning $40 shoveling snow.

A -40B 40C 0D +/- 40

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You dive 35 feet to explore a sunken ship.

A -35B 35C 0D +/- 35

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10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

The numbers -4 and 4 are shown on the number line.

Both numbers are 4 units from 0, but 4 is to the right of 0 and -4 is to the left of zero.

The numbers -4 and 4 are opposites. Opposites are the same distance from zero.

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7 What is the opposite of -7?

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8 What is the opposite of 18?

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Integers are used in game shows.

In the game of Jeopardy you:· gain points for a correct response· lose points for an incorrect response· and can have a positive or negative score

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When a contestant gets a $200 question correct: Score = $200Then a $100 question incorrect: Score = $100Then a $300 question incorrect: Score = - $200

How did the score become negative?

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9 After the following 3 responses what would the contestants score be?

$100 incorrect $200 correct $50 incorrect

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$200 correct$50 correct $300 incorrect

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$150 incorrect$50 correct $100 correct

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· An integer is a natural number, zero or its opposite.

· Number lines have negative numbers to the left of zero and positive numbers to the right.

· Zero is neither positive nor negative

· Integers can represent real life situations

To Review

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Absolute Value


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Absolute Value of Integers

The absolute value is the distance a number is from zero on the number line, regardless of direction.

Distance and absolute value are always non-negative (positive or zero).

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

What is the distance from 0 to 5?

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10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

What is the distance from 0 to -5?

Absolute Value of IntegersThe absolute value is the distance a number is from zero on the number line, regardless of direction.

Distance and absolute value are always non-negative.

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10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Absolute value is symbolized by two vertical bars

4

What is the 4 ?

This is read, "the absolute value of 4"

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-4 = 4

-9 = 9

= 99

Use the number line to find absolute value.

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

Moveto

check

Move to

check

Moveto

check

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-712 Find

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-2813 Find

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5614 What is ?

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-815 Find

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316 Find

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17 What is the absolute value of the number shown in the generator?

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18 Which numbers have 15 as their absolute value?

A -30B -15C 0D 15

E 30

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19 Which numbers have 100 as their absolute value?

A -100B -50C 0

D 50E 100

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Comparing and Ordering Integers


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Comparing Positive IntegersAn integer can be equal to; less than ; or greater than another integer.

The symbols that we use are:

Equals "=" Less than "<" Greater than ">"

For example:

4 = 4 4 < 6 4 > 2

When using < or >, remember that the smaller side points at the smaller number.

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20 The integer 8 is ______ 9.

A =B <C >

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A =B <C >

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A =B <C >

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To compare integers, plot points on the number line.

The numbers farther to the right are greater.

The numbers farther to the left are lesser.

Use the Number Line

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

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Place the number tiles in the correct places on the number line.

4

-45

-3

-2

30

2 -5

-1

1

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4-4 5-3 -2 30 2-5 -1 1

Now, can you see:

Which integer is greater?

Which is lesser?

click to reveal

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Put these integers on the number line.

-3 -1 0

5

4

-2 -72 -5

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-3 -1 0 54-2-7 2-5

Now, can you see:

Which integer is greater?

Which is lesser?

click to reveal

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Comparing Negative Integers

The greater the absolute value of a negative integer, the smaller the integer. That's because it is farther from zero, but in the negative direction.

For example:

-4 = -4 -4 > -6 -4 < -2

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

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Comparing Negative Integers

One way to think of this is in terms of money. You'd rather have $20 than $10.

But you'd rather owe someone $10 than $20.

Owing money can be thought of as having a negative amount of money, since you need to get that much money back just to get to zero.

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23 The integer -4 is ______ -4.

A =B <C >

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

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24 The integer -4 is ______ -5.

A =B <C >

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

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25 The integer -20 is ______ -14.

A =B <C >

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26 The integer -14 is ______ -6 .

A =B <C >

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Comparing All Integers

Any positive number is greater than zero and any negative number.

Any negative number is less than zero and any positive number.

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27 The integer -4 is ______ 6.

A =B <C >

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28 The integer -3 is ______ 0.

A =B <C >

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A =B <C >

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30 The integer -4 is ______ -9.

A =B <C >

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31 The integer 1 is ______ -54.

A =B <C >

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32 The integer -480 is ______ 0.

A =B <C >

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Drag the appropriate inequality symbol between the following pairs of integers:

1) -3 5

3) 63 36

5) -6 -3

7) -24 -17

9) 8 _____ -8

2) -237 -259

4) -10 -15

6) 127 172

8) -2 -8

10) -10 _____ -7

< >

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A thermometer can be thought of as a vertical number line. Positive numbers are above zero and negative numbers are below zero.

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33 If the temperature reading on a thermometer is 10℃, what will the new reading be if the temperature:

falls 3 degrees?

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rises 5 degrees?

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falls 12 degrees?

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36 If the temperature reading on a thermometer is -3℃, what will the new reading be if the temperature:

falls 3 degrees?

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rises 5 degrees?

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falls 12 degrees?

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Integer Addition


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Symbols

We will use "+" to indicate addition and "-" for subtraction.

Parentheses will also be used to show things more clearly. For instance, if we want to add -3 to 4 we will write: 4 + (-3), which is clearer than 4 + -3.

Or if we want to subtract -4 from -5 we will write:-5 - (-4), which is clearer than -5 - -4.

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While this section is titled "Integer Addition" we're going to learn here how to both add and subtract integers using the number line.

Addition and subtraction are inverse operations (they have the opposite effect). If you add a number and then subtract the same number you haven't changed anything.

Addition undoes subtraction, and vice versa.

Integer Addition: A walk on the number line.

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1. Start at zero 2. Walk the number of steps indicated by the first number. -Go to the right for positive numbers -Go to the left for negative numbers3. Walk the number of steps given by the second number. 4. Look down, you're standing on the answer.

Integer Addition: A walk on the number line.

Here's how it works.

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Let's do 3 + 4 on the number line.

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

1. Start at zero 2. Walk the number of steps indicated by the first number.3. Take the number of steps given by the second number. 4. Look down, you're standing on the answer.

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10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10


· Go to the right for positive numbers

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10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10



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Let's do -4 + (-5) on the number line.

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10


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10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10



· Go to the left for negative numbers

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10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10




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Let's do -4 + 9 on the number line.

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10


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10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10




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10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10




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Let's do 5 + (-7) on the number line.

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10


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10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10




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10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10




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Integer Addition: Using Absolute Values

You can always add using the number line.

But if we study our results, we can see how to get the same answers without having to draw the number line.

We'll get the same answers, but more easily.

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10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

3 + 4 = 7

-4 + 9 = 5

5 + (-7) = -2

-4 + (-5) = -9 10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

We can see some patterns here that allow us to create rules to get these answers without drawing.


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To add integers with the same sign 1. Add the absolute value of the integers. 2. The sign stays the same.

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-103 + 4 = 7

-4 + (-5) = -9 10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

3 + 4 = 7; both signs are positive; so 3 + 4 = 7

4 + 5 = 9; both signs are negative; so -4 + (-5) = -9


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Interpreting the Absolute Value Approach

The reason the absolute value approach works, if the signs of the integers are the same, is:

The absolute value is the distance you travel in a direction, positive or negative.

If both numbers have the same sign, the distances will add together, since they're both asking you to travel in the same direction.

If you walk one mile west and then two miles west, you'll be three miles west of where you started.

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To add integers with different signs 1. Find the difference of the absolute values of the integers. 2. Keep the sign of the integer with the greater absolute value.

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10-4 + 9 = 5

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-105 + (-7) = -2

9 - 4 = 5; 9 > 4, and 9 is positive; so -4 + 9 = 5

7 - 5 = 2; 7 > 5 and 9 is negative; so 5 + (-7) = -2


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Interpreting the Absolute Value Approach

If the signs of the integers are different:

For the 2nd leg of your trip you're traveling in the opposite direction of the 1st leg, undoing some of your original travel. The total distance you are from your starting point will be the difference between the two distances.

The sign of the answer must be the same as that of the larger number, since that's the direction you traveled farther.

If you walk one mile west and then two miles east, you'll be one mile east of where you started.

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3911 + (-4) =

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40-9 + (-2) =

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415 + (-8) =

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424 + (12) =

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43-14 + 7 =

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44-4 + (-4) =

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45-5 + 10 =

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Turning Subtraction Into Addition


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Subtracting IntegersSubtracting a number is the same as adding it's opposite.

We can see this from the number line, remembering our rules for directions. Compare these two problems: 8 - 5 and 8 + (-5).

For "8 - 5" we move 8 steps to the right and then move 5 steps to the left, since the negative sign tells us to move in the opposite direction that we would for +5.

For "8 + (-5)" we move 8 steps to the right, and then 5 steps to the left since we're adding -5.

Either way, we end up at +3.

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

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Subtracting Negative IntegersCompare these two problems: 8 - (-2) and 8 + 2.

For "8 - (-2)" we move 8 steps to the right and then move 2 steps to the right, since the negative sign tells us to move in the opposite direction that we would for -2.

For "8 + 2" we move 8 steps to the right, and then 2 steps to the right since we're adding 2.

Either way, we end up at +10.

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10

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Subtracting IntegersAny subtraction can be turned into addition by:

· Changing the subtraction sign to addition.

· Changing the integer after the subtraction sign to its opposite.

EXAMPLES:

5 - (-3) is the same as 5 + 3

-12 - 17 is the same as -12 + (-17)

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46 Convert the subtraction problem into an addition problem.

8 – 4

A -8 + 4B 8 + (-4)

C -8 + (-4)D 8 + 4

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-3 – (-10)

A -3 + 10

B 3 + (-10)

C -3 + (-10)

D 3 + 10

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-9 – 3

A -9 + 3

B 9 + (-3)

C -9 + (-3)

D 9 + 3

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6 – (-2)

A -6 + 2

B 6 + (-2)

C -6 + (-2)

D 6 + 2

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1 - 9 A -1 + 9

B 1 + (-9)

C -1 + (-9)

D 1 + 9

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Adding and Subtracting Integers

Review


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51 Calculate the sum or difference.

-6 – 2

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-10 + 6

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7 – (-4)

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4 – 7

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5 + (-5)

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9 + (-8)

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-4 + (-5)

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-2 – (-3)

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-2 + 4 + (-12)

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5 - 6 + (-7)

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-8 - (-3) + 5

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16 - (-9) - 21

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19 + (-12) - 11

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Multiplying Integers


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SymbolsIn the past, you may have used "x" to indicate multiplication. For example "3 times 4" would have been written as 3 x 4.

However, that will be a problem in the future since the letter "x" is used in Algebra as a variable.

There are two ways we will indicate multiplication: 3 times 4 will be written as either 3∙4 or 3(4).

Other commonly used symbols for multiplication include: * [ ] { } ( ) For example "3 times 4" could be written as: 3*4 3[4] 3{4} 3(4)

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Parentheses

The second method of showing multiplication, 3(4), is to put the second number in parentheses.

Parentheses have also been used for other purposes. When we want to add -3 to 4 we will write that as 4 + (-3), which is clearer than 4 + -3.

Also, whatever operation is in parentheses is done first. The way to write that we want to subtract 4 from 6 and then divide by 2 would be (6 - 4) / 2 = 1. Removing the parentheses would yield 6 - 4/2 = 4, since we work from left to right.

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Multiplication is just a quick way of writing multiple additions.

These are all equivalent:

3·43 +3 + 3 + 3 4 + 4 + 4

they all equal 12.

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We know how to add with a number line.

Let's just do the same thing with multiplication by just doing repeated addition.

To do that, we'll start at zero and then just keep adding: either 3+3+3+3 or 4+4+4.

We should get the same result either way, 12.

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10 2 3 4 5 6 7 8 9 10-1-2-3

Let's do 4 x 3 on the number line.

11 1312 14 16 1715

We'll do it as 3+3+3+3 and as 4+4+4

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10 2 3 4 5 6 7 8 9 10-1-2-3 11 1312 14 16 1715

Try 5 x 2 on the number line.

Try it as 5+5 and as 2+2+2+2+2

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Multiplying Negative Integers

Let's use the same approach to determine rules for multiplying negative integers.

If we have 4 x (-3) we know we can think of that as (-3) added to itself 4 times. But we don't know how to think of adding 4 to itself -3 times, so let's just get our answer this way:

4 x (-3) = (-3)+(-3)+(-3)+(-3)

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10 2 3-17 -1-2-3-4-5-6-7-8-9-10-11-13-15-16 -14 -12

4 x (-3) On the Number Line

4 x (-3) = (-3)+(-3)+(-3)+(-3)

So we can see that 4 x (-3) = -12

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4∙34 + 4 + 412

4(-3)(-3) + (-3) + (-3)-12

Multiplying Positive Integers has a positive value.

Multiplying a negative integer and a positive integer has a negative value.

What about multiplying together two negative integers: what is the sign of (-4)(-3)

Sign Rules for Multiplying Integers

?

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We can't add something to itself a negative number of times; we don't know what that means.

But we can think of our rule from earlier, where a (-) sign tells us to reverse direction.

10 2 3-17 -1-2-3-4-5-6-7-8-9-10-11-13-15-16 -14 -12

-5 - (-2) = -5 + 2

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So if we think of (-4)(-3) as -(4)(-3) we can then see that the answer will be the opposite of (-12): +12

Each negative sign makes us reverse direction once, so two multiplied together gets us back to the positive direction.

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4∙34 + 4 + 412

4(-3)(-4) + (-4) + (-4)-12

Multiplying Positive Integers yields a positive result.

Multiplying a negative integer and a positive integer yields a negative result.

Multiplying two negative integers together yields a positive result.

Sign Rules for Multiplying Integers

(-4)(-3)-((-4) + (-4) + (-4))-(-12)12

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Every time you multiply by a negative number you change the sign.

Multiplying with one negative number makes the answer negative.

Multiplying with a second negative change the answer back to positive.

1(-3) = -3 -3(-4) = 12


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When multiplying two numbers with the same sign (+ or -), the product is positive.

When multiplying two numbers with different signs, the product is negative.


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We can also see these rules when we look at the patterns below:

3(3) = 9 -5(3) = -153(2) = 6 -5(2) = -103(1) = 3 -5(1) = -53(0) = 0 -5(0) = 03(-1) = -3 -5(-1) = 53(-2) = -6 -5(-2) = 103(-3) = -9 -5(-3) = 15


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64 What is the value of 3∙7?

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65 What is the value of 5(-4)?

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66 What is the value of -3(-6)?

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67 What is the value of (-3)(-9)?

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68 What is the value of -8∙7?

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69 What is the value of -5(-9)?

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70 What is the value of 4(-2)(5)?

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71 What is the value of -2(-7)(-4)?

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72 What is the value of 6(3)(-8)?

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Dividing Integers


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Division Symbols

You may have mostly used the "÷" symbol to show division. Other common symbols include "/" and " "

We will also represent division as a fraction. Remember: 9 9÷3 = 33

are both ways to show division.

= 3

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Dividing IntegersDivision is the inverse operation of multiplication, just like subtraction is the inverse of addition.

When you divide an integer, by a number, you are finding out how many of that second number would have to add together to get the first number.

For instance, since 5∙2 = 10, that means that I could divide 10 into 5 2's, or 2 5's.

This is just what we did on the number line for multiplication, but backwards.

Let's try 10 ÷ 2

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10 2 3 4 5 6 7 8 9 10-1-2-3 11 1312 14 16 1715

Try 10 ÷ 2 on the number lineThis means how many lengths of 2 would be needed to add up to 10.

The answer is 5: the number of red arrows of length 2 that end to end give a total length of 10.

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10 2 3 4 5 6 7 8 9 10-1-2-3 11 1312 14 16 1715

Try 10 ÷ 5 on the number lineThis means how many lengths of 5 would be needed to add up to 10.

The answer is 2: the number of green arrows of length 5 that, end to end, give a total length of 10.

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10 2 3-17 -1-2-3-4-5-6-7-8-9-10-11-13-15-16 -14 -12

-12 ÷ 3 On the Number LineThis can be read as what would each step have to be if 3 of them was to take you to -12.

Each red arrow represents a step of 3, so we can see that -12 ÷ 3 = -4 (The answer is

negative because the steps are to the left.)

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-15 ÷ 3 = -5

We know that -5(3) = -15, so it makes sense that -15 ÷ 3 = -5.

We also know 3(-5) = -15. So, what is the value of -15 ÷ -5

The value must be positive 3, because 3(-5) = -15

-153 = -5

Dividing Integers

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The quotient of two positive integers is positive

The quotient of a positive and negative integer is negative.

The quotient of two negative integers is positive.

Dividing Integers

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73 Find the value of 32 ÷ 4

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74 Find the value of -4812

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75 Find the value of -27-9

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76 Find the value of -255

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77 Find the value of 0-13

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Powers of Integers


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Powers of IntegersPowers are a quick way to write repeated multiplication, just as multiplication was a quick way to write repeated addition.


24

2∙2∙2∙2 16

In this example 2 is raised to the 4th power. That means that 2 is multiplied by itself 4 times.

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Powers of IntegersBases and Exponents

When "raising a number to a power",

The number we start with is called the base, the number we raise it to is called the exponent.

The entire expression is called a power.

24

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When a number is written as a power, it is written in exponential form. If you multiply the base and simplify the answer, the number is now written in standard form.

EXAMPLE:

35 = 3(3)(3)(3)(3) = 243Power Expanded Notation Standard Form

TRY THESE:1. Write 53 in standard form.

2. Write 7(7)(7)(7)(7)(7)(7) as a power.

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78 What is the base in this expression? 32

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79 What is the exponent in this expression? 32

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81 What is the exponent in this expression? 43

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SquaresSquares - Raising a number to the power of 2 is called squaring it.

22 is two squared, and 4 is the square of 232 is three squared, and 9 is the square of 342 is four squared, and 16 is the square of 4

23 4

Area2 x 2 = 4 units2

Area =3 x 3 = 9 units2

Area =4 x 4 =

16 units2

34

2

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This comes from the fact that the area of a square whose sides have length 3 is 3x3 or 32 = 9;

The area of a square whose sides have length 5 is 5x5 or 52 = 25;

What would the area of a square with side lengths of 6 be?

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CubesCubes - Raising a number to the power of 3 is called cubing it.

23 is two cubed, and 8 is the cube of 233 is three cubed, and 27 is the cube of 343 is four cubed, and 64 is the cube of 4

That comes from the fact that the volume of a cube whose sides have length 3 is 3x3x3 or 33 = 27;

The volume of a cube whose sides have length 5 is 5x5x5 or 53 = 125;

etc.

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10 2 3 4 5 6 7 8 9 10-1-2-3

Let's do 22 on the number line.

11 1312 14 16 1715

22 = 2 x 2

Travel a distance of 2, twice

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10 2 3 4 5 6 7 8 9 10-1-2-3


11 1312 14 16 1715

32 = 3 x 3 = 3 + 3 + 3 = 9

Travel a distance of 3, three times

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10 2 3 4 5 6 7 8 9 10-1-2-3


11 1312 14 16 1715

23 = 2 x 2 x 2 23 = (2 x 2) x 2 First, travel a distance of 2, twice: 423 = 4 x 2 = 8 Then, travel a distance of 4, twice: 8

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10 2 3 4 5 6 7 8 9 10-1-2-3


11 1312 14 16 1715

42 = 4 x 4 = 4+4+4+4 = 16

Travel a distance of 4, four times

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10 2 3 4 5 6 7 8 9 10-1-2-3


11 1312 14 16 1715

24 = 2 x 2 x 2 x 224 = 2 x 2 x 2 x 2 First, travel a distance of 2, twice: 424 = 4 x 2 = 8 x 2 Then, travel a distance of 4, twice: 824 = 8 x 2 = 16 Then, travel a distance of 8, twice: 16

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83 Evaluate 32.

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84 Evaluate 52.

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85 Evaluate 82.

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86 Evaluate 53.

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87 Evaluate 72.

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88 Evaluate 22.

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34

3∙3∙3∙3 81

If we now take(-3)4

(-3)(-3)(-3)(-3)81

In this example we still get 81. Why?

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If we take (-3)3

It is equivalent to(-3)(-3)(-3)

Which equals-27

Why is this example negative?

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89 Evaluate (-5)2.

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90 Evalute (-2)3.

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91 Evaluate (-4)4.

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92 Evaluate (-7)2.

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93 Evaluate (-3)5.

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Remember...these are all equivalent: 34

3∙3∙3∙3 81In both examples we get 81.

If we take - 34

We get -81.

This answer is negative because - 34 means the opposite of 34.

(-3)4

(-3)(-3)(-3)(-3)81

Note: When there are no parentheses, we take the exponent first and then apply the negative sign!

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94 Evaluate -52.

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95 Evalute -23.

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96 Evaluate -44.

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97 Evaluate -72.

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98 Evaluate -35.

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Rules for Exponents


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page145svg

Exponent Rules - Multiplying

You can only directly multiply numbers which have exponents if the bases are the same. If the bases are the same, just add the exponents.

(43)(42) = 4(3+2) = 45

Here's why: (43)(42) = (4x4x4)(4x4) = 4x4x4x4x4 = 45

This will be true for any base, not just 4, so this rule always works.

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99(54)(53) = 5?

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100(43)(42) = 4?

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101(37)(31) = 3?

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102(24)(2-2) = 2?

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103(158)(153) = 15?

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104(64)(6-8) = 6?

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105 (43)(42)(45) = 4?

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106 (24)(25)(27) = 2?

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107 (64)(6?) = 610

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108(57)(5?) = 53

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Exponent Rules - DividingYou can only directly divide numbers which have exponents if the bases are the same.

If the bases are the same, just subtract the exponent of the denominator from that of the numerator.

75 = 7x7x7x7x7 = 72

73 = 7x7x7

This will be true for any base, not just 7, this rule always works.

75 ÷ 73 = 7(5-3) = 72

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10954 ÷ 53 = 5?

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11074 ÷ 73 = 7?

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11193 ÷ 96 = 9?

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11234 ÷ 38 = 3?

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113814 ÷ 86 = 8?

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11484 ÷ 84 = 8?

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115 (55)(53)÷ 52 = 5?

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116 98 ÷ 9? = 95

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117 66 ÷ 6? = 613

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118 4? ÷ 48 = 43

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Exponent Rules - Exponent of zeroANY non-zero base raised to the zero power is equal to one.

Consider the following pattern of powers:

23 = 8

22 = 4

21 = 2

20 = 1

_ 2_ 2

_ 2

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11980 = ?

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12040 = ?

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121(-5)0 = ?

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122250 = ?

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Exponent Rules - Power of a powerWhat happens when a power is raised to a power?

Let's see...

(34)2 = 34(34) = [3(3)(3)(3)][3(3)(3)(3)] = 38

(a3)4 = (a3)(a3)(a3)(a3) = a(a)(a)(a)(a)(a)(a)(a)(a)(a)(a)(a) = a12

What do you notice?

Simplify: (52)6

Move the box to see the rule... (xm)n = xmn

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123(38)3 = 3?

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124(139)12 = 13?

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125(28)5 = 2?

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126(114)8 = 11?

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Exponent Rules - Negative Exponents

Consider the following pattern of powers:

23 = 8

22 = 4

21 = 2

20 = 1

2-1= 2-2=

2-3=

_ 2_ 2

_ 2_ 2_ 2

_ 2

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Negative Exponents

Considering the pattern of powers for any nonzero number:

5-2 = Then, = 52

= 1 _ = 25

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1272-1 = ?

Give your answer as a fraction

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128

= ? 14-1

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12910-2 = ?

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1305-1 = ?

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131

= ? 1

3-3

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1322-5 = ?

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133

= ? 15-2

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134

= ? 19-2

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integers - njctlcontent.njctl.org/courses/math/archived-coursesunits-prior-to-comm… · integers...

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