1

Exponents

Exponents tell us how many times to multiply a base number by itself.

Exponential form: 54 exponent

base number

Expanded form: 5•5•5•5

25•5•5

125•5

625

To use a calculator: put in the base number, push the exponent button (yx

or ∧); enter the exponent; push equal (=) button.

*Anything to the power of zero equals 1.

Ex) 250=1

*Anything to the first power (power of one) equals the base number.

Ex) 251=25

2

Square Roots and Perfect Squares

Definition: a square root of any number is the number that when multiplied by itself

(to the power of 2 or squared) gives you the number under the square root symbol.

The exponent used with square roots is 2.

Square root symbol:

81 =9 36 =6 16 =4 49 =7 For example, the number 7

multiplied by itself equals 49 or the number under the square root symbol (7•7=49).

The number under the square root symbol is the area of a square. The answer is the

length of each side of the square (even if the answer is a decimal)

3 10 6

6

3 10

9 =3 100 =10 36 =6

Perfect squares are square root problems with the answer (the sides of the square)

being a whole number (no decimals). SOL NOTE: You must remember all perfect

squares up to 400 =20

0 =0 1 =1 4 =2 9 =3 16 =4 25 =5 36 =6

49 =7 64 =8 81=9 100 =10 121 =11 144 =12 169 =13

196 =14 225 =15 256 =16 289 =17 324 =18 361 =19 400 =20

If the square root problem is not a perfect square, use perfect squares to see the

number above and below to estimate.

Ex) The square root of 32 ( 32 ) will be between 5 and 6. The square root of 25 =5

and the square root of 36 =6. Since 32 is between 25 and 36, we know the answer

of 32 will be between 5 and 6 and will actually be closer to 6 than 5.

3

Base Ten Exponents

Place values:

Ten thousands 10,000. 104

Thousands 1,000. 103

Hundreds 100. 102

Tens 10. 101

Ones 1. 100

Tenths .1 10-1

Hundredths .01 10-2

Thousandths .001 10-3

Ten thousandths .0001 10-4

*With positive base ten exponents, the exponent will match the zeros in the answer.

*With negative base ten exponents the negative exponent will match the number of digits in

the answer to the right of the decimal.

Equivalent numbers:

10-5

= 1•10-5

= .00001 = 1 = 1 = 1

105

10•10•10•10•10 100,000

*If you don’t memorize the above chart or the rules for the SOL, this is another method. First

take the negative exponent and put it into scientific notation by just putting 1 x in front of the

base 10 exponent ex)10-5

=1. X 10-5

Now, follow rules for scientific notation to make this a

decimal. Because this is a negative exponent the decimal should be moved 5 place values to the

left. (5 digits left). Therefore, 4 zeros need to be added before the one. 0.00001. Now read the

decimal as “one hundred thousandths” and put this in fraction form.

1 Which is also 1 or 1

100,000 105

10•10•10•10•10

Reciprocal- flip/flop: 10-3

is the reciprocal of 103

ten

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3 6 , 1 2 3 . 1 3 7 2 4

4

Scientific Notation

Scientific notation is a way to write extremely large or small numbers with a lot of digits as an

abbreviation using base 10 exponents.

Step 1 – from your original number, make a number between 1-10 with a decimal

Step 2 – count the number of digits or place values between original position of the decimal and

where you moved the decimal

Step 3 – use the number you get when counting from step 2 as the base 10 exponent

If the original number is greater than 1, the exponent is positive. If the original number is

between 0 and 1, the exponent is negative

Ex) 579,000

step 1: 5.79

step 2: 579,000. 5 places

step 3: 579,000 = 5.79 x 105

ex) 6,319,000,000. = 6.319 x 109

Common mistakes

1) 8,017,000,000 ≠ 8.017 x 106 - don’t count just the zeros, count the place values the

decimal has moved or the number of digits between the original decimal and the new.(correct

answer is 8.017 x 109)

2) 8,017,000,000 ≠ 8.017 x 1010

– don’t count all the place values, only count the place values

you moved the decimal

3) 4,003,000,000 ≠ 4.3 x 108 – you cannot get rid of “number locked zeros” (correct answer is

4.003 x 109)

Putting numbers back in to standard form

Step 1: rewrite the first part of the number without the decimal

Step 2: look at the exponent and add zeros to get to the place value needed

5.79 x 105

Step 1: 579

Step 2: 579,000. Needed to move the decimal 5 spaces, had moved 2 so needed to add 3 zeros.

Common mistakes

1) Do not add too many zeros – exponents tell place values (CM 57,900,000)

2) Do not forget to move the decimal (CM 5.79000)

5

Scientific Notation with Negative Exponents

The number will be greater than 0 and less than one.

Step 1 – from your original number, make a number between 1-10 with a decimal.

Step 2 – count the number of digits or place values between original position of the

decimal and where you moved the decimal

Step 3 – use above number as the base 10 exponent (if the original number is less

than one, the exponent must be a negative)

Ex) 0.000000003165

step 1: 3.165

step 2: 0.000000003165 9 places (9 digits)

step 3: 0.000000003165 = 3.165 x 10-9

ex) 0.00038 = 3.8 x 10-4

0.0000006 = 6 x 10-7

4.013 x 10-5

- .00004013

*When going from standard form back into scientific notation form, count

the digits between where the decimal is in standard form to where it will

be when making a number between one and ten. The number of digits

gives us the exponent.

*Remember to make the exponent a negative if the original number in

standard form is between zero and one.

6

Fractions, Decimals and Percents

4

1 = 25% = 0.25 These are all equivalent but not necessarily equal. They represent the same

amount of a whole, but to be equal, the whole must be the same.

Ex) Comparing a large pizza to a small pizza - �

� of the large pizza is bigger than

�

� of the small

pizza but they are equivalent amounts.

Fractions, decimals and percents are three different ways to represent the same part to whole

relationships.

When is it whole? Part Whole Percent the whole is 100 25% out of 100% Decimal the whole is 1 .25 out of 1 Fraction the whole is the denominator (The division symbol (÷)

equals atordeno

numerator

min so all fractions can be considered

division problems.)

�

� out of

�

�

Changing a decimal to a percent – multiply by 100 because the whole is 100 times 1

(whole in decimal is 1; whole in percent is 100). This can be done by moving the

decimal 2 places to the right. You can still have decimals in percent form.

Ex) .67 =?% - 0.67 • 100 = 67 so 0.67 = 67%

.0305 • 100 = 3.05%

Changing a percent to a decimal – divide by 100 (move decimal 2 places to the left.

Ex) 67% = ?% - 67 ÷ 100 = 0.67 so 67% = 0.67

.67 is 100 times smaller than 67, just like 1 is 100 times smaller than 100.

Changing a fraction to a decimal – divide the numerator by the denominator.

Ex) 4

1 = ? - 14 = 0.25 so

4

1 = 0.25 (Numerator arrives 1

st to the hotel so it gets the last

room) Ex) 2

1= 1÷2 = 2 1 Common mistake 1 2

Changing a fraction to a percent – first change the fraction to a decimal and then

follow decimal to percent rule (multiply by 100)

Ex) 4

1 = 0.25 = 25%

7

Converting Percents to Fractions

When converting a percent to a fraction, when the percent is a whole number, just

put the percent number over 100 since 100 is the whole number for percents. You

can then simplify the fraction (find the greatest common factor to simplify).

Ex) 62% = 100

62 simplified =

50

31

If there is a decimal in the percent, convert the percent to the decimal form then put

it into a fraction by reading then writing.

Ex) 71.4% = 0.714 which reads “seven hundred fourteen thousandths” which written

is 1000

714 and simplified is

500

357.

Conversion chart (Memorize for SOL)*This is on the non calculator section.*

Fraction (whole = denominator) Decimal (whole =1) Percent (whole = 100)

numeratoratordenomin Move decimal 2 spaces to the right (x100)

Say it correctly, then write it Move decimal 2 spaces to the left (÷100)

8

Repeating Decimals

Line over a number in a decimal shows that it repeats. Only the number under the

line is repeated.

Ex) 0. 4� = 0.4444444…….

0. 312����� = 0.312312312312…..

0.156� = 0.5166666…..

Common mistake – comparing decimals to determine which is greater/less. Must

line up the matching place values. (Some students say the repeating decimal is

greater because it never ends but . 7� is not greater than .8)

Ex) Which is greater? 0. 31���� or 0.7 It would appear that 31 is bigger than 7 but when

you line them up by the place value you see that 0.7 > 0. 31����

0.31313131…

0.70000000…

To change a repeating decimal to a percent, round it and show that it is approximate.

Ex) 0. 31���� ≈ 31.3%

9

Ordering Fractions, Decimals Percents and Scientific Notation

*When solving an ordering problem, make sure you read if it is least to greatest or

greatest to least.

If the numbers are all in scientific notation and you are ordering them, you only need

to look at the exponent. The bigger the exponent the bigger the number.

Ex) 3•1010

, 7.8•104, 15•10

3 in order from least to greatest would be:

15•103, 7.8•10

4, 3•10

10

Most of the time you put all the numbers into decimal or percent form to compare

NOT fractions. Decimal form is used most frequently. Decimal form for scientific

notation is standard form. The numbers can then be stacked to compare and order.

(Create a place value chart.)

Ex) Place the following in order from greatest to least 71%, 10

7, .007, 0.71�, 7.1•10

-3,

7.1•106

*Once you line up your place values, put zeros in to make all numbers have the same

place value. (Unless it is repeating, the you put in whatever number repeats instead

of zero.)

71% 0.7100

10

7 0.7000

.007 0.0070

0.71� 0.71�11 (CM - putting in zeros here instead of ones)

7.1x10-3

0.0071

7.1x106

7,100,000.0000

Greatest to least: 7.1•106, 0.71�, 71%,

10

7, .007, 7.1•10

-3

10

Integers

Integers are the set of all whole numbers and their opposites. (…

-3,

-2,

-1, 0, 1, 2,

3…) Any number with a fraction, decimal or percent is NOT an integer. (-3.2,

-1.5,

75% , 10

7..015)

Read these symbols correctly from left to right:

> Greater than

< Less than

The negative symbol can be read as “the opposite of.”

Common mistake – integers are often confused with just negative numbers.

*Remember that integers also include positives.

-1 1 2 3 4 5

-4

-3

-2 0

-5

-6

11

Basic Rules for Integers – MEMORIZE!

Addition

• Signs are the same

Add and keep signs.

Ex) -3 +

-5=

-8

3 + 5=8

• Signs are different

Ask yourself two questions:

a) “Do I have more negative numbers or positives?” (This gives us the sign for the answer)

b) “How many more negatives or positives do I have?” (This gives us the number for our

answer)

Ex) -8 + 3=

?

a) More negatives or positives? Negatives so my answer is a negative number.

b) How many more? 5

So the answer is negative five -8 + 3=

-5

Subtraction

Change subtraction to addition and change the sign of the second number. (When subtracting,

just add the opposite.)

Ex) 6-(-3) = 6+

+3=9

Next, just follow addition rules.

Ex) -6-(

-3) =

-6+

+3= Follow addition rules for one negative and one positive. I have 3 more

negatives so my answer is negative 3. -6-(

-3) =

-6+

+3=

-3

Multiplication and Division

When multiplying or dividing, when the signs are the same the answer is positive.

Ex) 3•6=18 -3•

-6=18

4÷2=2 -4÷

-2=2

When multiplying or dividing, when the signs are different the answer is always negative.

Ex) 3•(-6)=(

-18)

-18÷6=(

-3)

12

Integer Addition

Integer addition when you have all negatives or all positives: just add the numbers

and keep the sign the same. (positive plus positive equals positive; negative plus

negative equals negative)

Ex) 3 + 4=7 (positive plus positive equals positive) -3 +

-4=

-7 (negative plus negative equals negative)

Zero pairs – equal negative and positive numbers that add together to be zero. (-3, 3;

-5, 5)

Addition with negative and positive, look for zero pairs. The smaller of the two

numbers you are adding is the number of zero pairs. Whatever you have left is the

answer. Do you have more negatives or positives? How much more?

Key:

= positive = negative

Ex) -3+8=5

Zero pairs cancel each other out (add together to make zero)

5 positive chips left

10+-6=4

13

Addition of Integers with Chips

Whichever you have less of (negative or positive) is the number of zero pairs you

have.

72+-103 there are 72 zero pairs (31 negative chips left over so your answer is -31)

-8+3 there are 3 zero pairs (5 negative chips left over so your answer is -5)

It is not which number is the least, it is which you have the less of (negatives or

positives).

Be sure to look at the chip key.

*We will use this key for example.

Key: or Key:

= positive = negative = positive = negative

Ex) -5+

-2 I have 5 negative chips and need to add 2 negative chips. The answer is

-7

No zero pairs.

3+4 I have 3 positive chips and need to add 4 positive chips. The answer is 7.

No zero pairs.

-6+4 I have 6 negative chips and need to add 4 positive chips. The answer is -2.

There are 4 zero pairs with 2 negative chips left over.

Using mailman math to explain zero pairs: I received a bill for $8. I received a check

for $3. I can use the $3 to pay for 3 of the 8 I owe. I still owe $5.

14

Integer Addition with Number Line Model

Number lines extend forever in both directions. You can show any portion of the line as needed

by a problem.

You always start with zero when using the number line model.

If the number is negative go to the left.

If the number is positive go to the right.

Overlapping lines are the zero pairs.

Ex) -3+

-2= start at zero, go to the left to

-3 then go 2 more to the left. Answer is

-5.

Ex) 3+-7= start at zero, go to the right to 3 the go to the left 7. Answer is

-4.

-1 1 2 3 4 5

-4

-3

-2 0

-5

-6

-1 1 2 3 4 5

-4

-3

-2 0

-5

-6

Zero pairs

15

Rules for Subtracting Integers

Rule: Change your subtraction sign to an addition sign and change the second

number to its opposite. Then follow the addition rules.

Any time you subtract, you are adding the opposite of the second number.

Two possible ways to have a gain using mail man math problems: 1) receive a check

(adding a positive) or 2) have a bill taken away (subtracting a negative)

Ex) (-7)-6=

-13 I had a bill for $7 then lost a check for $6. I am in the hole 13 dollars.

4-(-3)=7 I received a check for $4 and a bill was taken away for $3. I have gained 7

dollars.

Or, just follow the rules:

(-7)-6 = (

-7)+(

-6) =

-13

4-(-3)=7

16

Integer Chip Subtraction

Always show the first number in chips first. See if you can take away the second

number from those chips. If not, add as many zero pairs as needed. (Zero pairs do

not change the first number you are just adding zero to it.) Use the basic integer

subtraction rule to check your work.

Key:

= positive = negative

Ex) -10-(

-6) I have 10 negative chips and need to take away 6 negative chips.

The answer is -4.

Check: -10-(

-6) =

-10+6=

-4

In this problem, it was easy to take the second amount of chips from the first.

Ex) 5-(-6) I have 5 positive chips and need to take away 6 negative chips.

Since there are no negative chips in 5, add enough zero pairs ( ) to

be able to cross out 6 negative chips (6 zero pairs are needed). Now

you can subtract the 6 negative chips and you are left with 11 positive

chips.

5-(-6)=11. Check: 5-(

-6)=5+6=11.

17

Integer Subtraction with the Number Line Model

Always start at zero. Go to your first number. If you are taking away a negative, go right. If you

are taking away a positive, go left. Number line model can be the same for addition and

subtraction.

Ex) -6-(

-4) =

-2

-6+4 =

-2

In this problem, we went 4 spaces to the right because we were taking away negatives. (We

would be getting a bigger number.

A way to check if your subtraction model is correct is to change the subtraction to an addition

problem. Check to see if your subtraction model and addition model would be the same.

Balloon Math for adding and subtracting integers

Sand bags are weight so you travel down (-)

Gas bags are air so you travel up (+)

Start at zero. Travel to first number. Take away either

sand bags (-) or gas bags (+)

Ex. -3-(

-4) Start at zero. Travel to

-3. Take away 4 sand bags.

End up at 1.

*Ways for a balloon to go up

• Adding gas bags (plus positive)

• Taking away sand bags (subtracting a negative)

*Ways for a balloon to go down

• Adding sand bags (weight) (plus a negative)

• Subtracting gas bags (taking away a positive)

-1 1 2 3 4 5

-4

-3

-2 0

-5

-6

0

4

3

2

1

-1

-4

-3

-2

18

Multiplication and Division of Integers

The rules for the multiplication and division of integers are the same.

Positive x positive = positive 2•7=14 positive ÷ positive = positive 14÷7=2

Negative x negative = positive -2•

-7=14 positive ÷ negative = negative 14÷

-7=

-2

Negative x positive = negative -2•7=

-14 negative ÷ positive = negative

-14÷7=

-2

When multiplying or dividing, if the signs are the same, the answer is positive. If the

signs are different, the answer is negative.

The first number tells how many groups. The second number tells how many are in

each group.

3•2 means there are 3 groups of 2 = 6

Using balloon math: The sign of the first number tells whether we are putting on

groups (+) or taking off groups (-). The second number tell whether they are gas bags

(+) or sand bags (-)

-2•7=

-14 You are taking away 2 groups of 7 gas bags, so you are going down 14

spaces.

2•-7=

-14 You are adding 2 groups of 7 sand bags, so you are going down 14 spaces.

-2•

-7=14 You are taking away 2 groups of 7 sand bags, so you are going ups 14

spaces.

19

Integer Word Problems

Examples are in handout package.

Usage Positive Negative

Bank accounts Deposit/credit Withdrawal/debit

Stocks Gain Loss

Temperature Above Below

Altitude Ascend/above sea level Descend/below sea level

Football Gaining yardage Losing yardage/penalty

Business Profit Loss

Golf-negatives are good,

positives are bad; par is 0

or even

Over par Under par

Pictures are a must. You must draw pictures

Any time you are doing addition or subtraction word problems, the picture will be a vertical

number line. Labels are important. You must have a completion sentence that states the answer.

Multiplication and division word problems are most likely not number lines.

Ex) Grace recorded the average temperature outside for three days.

Monday 19°F

Tuesday -10°F

Wednesday 7°F

What is the difference in temperature between Monday and Tuesday? The difference is 29°F.

0°

19° Monday’s temp.

-10° Tuesday’s temp.

19 spaces{ 10 spaces{

20

Order of Operations with Integers

*Order of operation problems are simplifying expressions. Do in this order:

When students use Please Excuse My Dear Aunt Sally (PEMDAS) to remember the order of

operations, they often forget that the order of M/D and A/S can switch because they occur left to

right whichever comes first.

The directions to an order of operations problem will say to “Evaluate” or “Simplify.” It won’t say

to solve because it doesn’t have an equal sign.

Any time there are a lot of operations inside parenthesis, it must be treated like its own order of

operations problem. Parenthesis without operations inside, means multiply.

Ex)

8•22+(2+3)-10÷2

8•22+5-10÷2

8•4+5-10÷2

32+5-10÷2

32+5-5

37-5

32

3(32÷4.5•2

3)

2+7•5

2

3(9÷4.5•23)

2+7•5

2

3(9÷4.5•8)2+7•5

2

3(2•8)2+7•5

2

3(16)2+7•5

2

3(256)+7•52

3(256)+7•25

768+7•25

768+125

943

*Many common mistakes are made when simplifying expressions. Do one step at a time and be

careful with exponents.

*With negative exponents remember if the exponent is even the answer will be positive. If the

exponent is odd the answer will be negative.

Ex) (-3)

2=

-3•

-3=9

Ex) (-3)

3=

-3•

-3•

-3

9•-3=

-27

parenthesis exponents

Multiplication or division

(left to right whichever comes first) Addition or subtraction

(left to right whichever comes first)

21

With fraction problems, do numerator and denominator as separate “mini” problems.

82÷-4

(31+1) ÷-2

Numerator

82÷-4

64÷-4

-16

Denominator

(31+1) ÷-2

(3+1) ÷-2

4÷-2

-2

2

16

−

−= 8

22

Proportions and Ratios

Ratio – the comparison of two numbers by division (similar to fraction). Fractions are

always ratios but ratios are not always fractions. A comparison of any two quantities

which may be expressed…3

2, 2:3, 2 to 3, 2 out of 3

Proportion simply shows us that we have two equivalent ratios (not necessarily

equal).

Proportional reasoning – the ratios will increase or decrease by the same amount of

times either numerator to numerator and denominator to denominator or both

numerators to denominators.

If two ratios can be reduced to the same fraction, they are proportional (equivalent).

Ex) Are 10

5

and 12

6

proportions? Yes because 5•2=10 and 6•2=12 (same increase

from numerator to denominator on both ratios).

Ex) Are 6

2

and 126

42

proportions? Yes because 2•21=42 and 6•21=126 (same

increase numerator to numerator and denominator to denominator). Or yes because

2•3=6 and 42•3=126.

Cross multiplication is another way to check for proportion.

Ex)5

2

5.17

7

5•7=35 and 2•17.5=35 so the two ratios are proportional.

If two ratios are proportional and there is a variable, use proportion reasoning or

cross multiply to find the variable.

Ex) 6

2

=n

42

Using proportional reasoning, 2•3=6 so 42•3=n so n is 126.

23

Solving Proportions Using Algebra

You must always keep your equation balanced. Whatever you do to one side you

must do the same to the other.

Step one – cross multiply (always keep the variable on the left side of the equation)

Step two – 2 goals

1) Get the variable by itself (do the inverse operation to both sides)

2) Keep the equation balanced.

Ex) 12

5

=48

n

step one: (cross multiply) 12n=240. step two: (get n by itself) divide by 12. To keep

the equation balanced, 240 needs to be divided by 12 too. 12

12n

=12

240

n=20

Check: 12•20 = 240

24

Proportion Word Problems

You must label.

You must have three known variables to solve a simple proportion.

Step one: underline or highlight the three know values (including the label)

Step two: since proportions are two equivalent rations, look for the two values that

create the first ratio. (First two numbers being compared.)

Step three: Set up the second ration with the unknown value and make sure the

labels line up.

Step four: Solve your proportion and make your completion statement.

The label for the unknown value is the question being asked.

Some word problems have 3 labels but you need 2. Some have 1 label but you need

to have 2.

When looking for answers on a test that gives a proportion for answers, cross

multiply to see if it matches your answer.

Ex) 12 inches of rain fell in 1 hour. How many hours did it take to rain 36 inches?

Step one: underline the 3 known values.

Step two: set up first ratiohour

inches

1

12

Step three: set up second rationhours

inches36 with labels lined up

hour

inches

1

12

=nhours

inches36 solve with proportional reasoning 12•3=36 so n=1•3 (n=3)

Solve by cross multiplying 12n=36 Divide both sides by 12. n=3