The sat skill exam 5 pdf

Perfect Squares The square root of a number might not be a whole number.. A whole number is a perfect square if its square root is also a whole number.. Any time you see a number with a

Sometimes, you will see an exponent with a

vari-able: b n The “b” represents a number that will be a

fac-tor to itself “n” times.

Example:

b n where b = 5 and n = 3 Don’t let the variables

fool you Most expressions are very easy once you substi-tute in numbers

b n= 53= 5 × 5 × 5 = 125

Laws of Exponents

■ Any base to the zero power is always 1

Examples:

50= 1 700= 1 29,8740= 1

■ When multiplying identical bases, you add the

exponents

Examples:

22× 24× 26= 212 a2× a3× a5= a10

■ When dividing identical bases, you subtract the

exponents

Examples:

2 2

5 3

= 22 a a74= a3

Here is another method of illustrating multipli-cation and division of exponents:

b m × b n = b m + n

b

b

m n

= b m – n

■ If an exponent appears outside of the parentheses,

you multiply the exponents together

Examples:

(33)7= 321 (g4)3= g12

Squares and Square Roots

The square root of a number is the product of a

num-ber and itself For example, in the expression 32= 3 ×

3 = 9, the number 9 is the square of the number 3 If

we reverse the process, we can say that the number 3 is

the square root of the number 9 The symbol for square

root is 25 and it is called the radical The number inside of the radical is called the radicand.

Example:

52= 25; therefore,25 = 5

Since 25 is the square of 5, we also know that 5 is the square root of 25

Perfect Squares

The square root of a number might not be a whole number For example, the square root of 7 is 2.645751311 It is not possible to find a whole number that can be multiplied by itself to equal 7 A

whole number is a perfect square if its square root is

also a whole number

Examples of perfect squares:

1, 4, 9, 16, 36, 49, 64, 81, 100,

Properties of Square Root Radicals

■ The product of the square roots of two numbers

is the same as the square root of their product

Example:

a × b = a × b

5 × 3 = 15

■ The quotient of the square roots of two numbers

is the square root of the quotient

Example:

■ The square of a square root radical is the radicand

Example:

(N)2= N

(3)2= 3 × 3 = 9 = 3

√¯¯¯

√¯¯¯

a

√¯¯¯b

√¯¯¯5

=

b (b≠ 0)

√¯¯¯¯¯15

√¯¯¯3 √¯¯¯¯¯15

■ To combine square root radicals with the same radicands, combine their coefficients and keep the same radical factor You may add or subtract radicals with the same radicand

Example:

ab + cb = (a + c)b

43 + 23 = 63

■ Radicals cannot be combined using addition and subtraction

Example:

a + b  ≠ a + b

4 + 11 ≠ 4 + 11

■ To simplify a square root radical, write the radi-cand as the product of two factors, with one num-ber being the largest perfect square factor Then write the radical over each factor and simplify

Example:

8 = 4 × 2 = 22

Integer and Rational Exponents

Integer Exponents

When dealing with negative exponents, remember that

a –n= a1n .

Examples:

4–2=  4

1

2

=  1

1 6

 –2–3= –12 3= 

–

1 8

= –1

8 

Rational Exponents

Recall that rational numbers are all numbers that can

be written as fractions (23), terminating decimals (.75), and repeating decimals (.666 ) Keeping this in mind, it’s no surprise that numbers raised to rational exponents are just numbers raised to a fractional power

What is the value of 412?

412can be rewritten as 4, so it is equal to 2

Any time you see a number with a fractional exponent, the numerator of that exponent is the power you raise the number to, and the denominator is the root you take

Examples:

25 = 2

251



8 = 3

81



16 = 2

161



Divisibility and Factors

Like multiplication, division can be represented in a few different ways:

8 ÷ 3 38 8

3 

In each of the above, 3 is the divisor and 8 is the dividend

Odd and Even Numbers

An even number is a number that can be divided by the number 2: 2, 4, 6, 8, 10, 12, 14, An odd number

can-not be divided evenly by the number 2: 1, 3, 5, 7, 9, 11,

13, The even and odd numbers listed are also exam-ples of consecutive even numbers and consecutive odd numbers because they differ by two

Here are some helpful rules for how even and odd numbers behave when added or multiplied:

even + even = even and even × even = even odd + odd = even and odd × odd = odd odd + even = odd and even × odd = even

Dividing by Zero

Dividing by zero is not possible This is important to remember when solving for a variable in the denomi-nator of a fraction

Example:



a –

6 3



In this problem, we know that a cannot be equal to

3, because that would yield a zero in the denominator

a – 3 = 0

a≠ 3

12

1

12

Factors and Multiples

Factors are numbers that can be divided into a larger

number without a remainder

Example:

12 ÷ 3 = 4 The number 3 is, therefore, a factor

of the number 12 Other factors of

12 are 1, 2, 4, 6, and 12

The common factor of two numbers are the

fac-tors that both numbers have in common

Example:

The factors of 24 = 1, 2, 3, 4, 6, 8, 12, and 24

The factors of 18 = 1, 2, 3, 6, 9, and 18

From the above, you can see that the common factors of 24 and 18 are 1, 2, 3, and 6 From this list, we

can also determine that the greatest common factor of

24 and 18 is 6 Determining the greatest common

fac-tor is useful for reducing fractions

Any number that can be obtained by multiplying a

number x by a positive integer is called a multiple of x.

Example:

Some multiples of 5 are: 5, 10, 15, 20, 25, 30, 35,

40 Some multiples of 7 are: 7, 14, 21, 28, 35, 42, 49,

56

From the above, you can also determine that the

least common multiple of the numbers 5 and 7 is 35.

The least common multiple, or LCM, is used when

performing various operations with fractions

Prime and Composite Numbers

A positive integer that is greater than the number 1 is

either prime or composite, but not both

■ A prime number has only itself and the number 1

as factors

Examples:

2, 3, 5, 7, 11, 13, 17, 19, 23,

■ A composite number is a number that has more

than two factors

Examples:

4, 6, 8, 9, 10, 12, 14, 15, 16,

■ The number 1 is neither prime nor composite

Prime Factorization

The SAT will ask you to combine several skills at once

One example of this, called prime factorization, is a

process of breaking down factors into prime numbers

Examples:

18 = 9 × 2 The number 9 can also be written

as 3 × 3 So, the prime factoriza-tion of 18 is:

18 = 3 × 3 × 2 This can also be demonstrated with the factors

6 and 3: 18 = 6 × 3 Because we know that 6 is equal to 2 × 3, we can write: 18 = 2 × 3 × 3

According to the commutative law, we know

that 3 × 3 × 2 = 2 × 3 × 3

Number Lines and Signed Numbers

You have surely dealt with number lines in your career

as a math student The concept of the number line is

simple: Less than is to the left and greater than is to the

right

Sometimes, however, it is easy to get confused about the value of negative numbers To keep things

simple, remember this rule: If a > b, then –b > –a Example:

If 7 > 5, then –5 > –7

–7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7

Greater Than

Less Than

Absolute Value The absolute value of a number or expression is always

positive because it is the distance a number is away from zero on a number line

Example:

1  1 2  4  2  2

Working with Integers

Multiplying and Dividing

Here are some rules for working with integers:

(+) × (+) = + (+) (+) = + (+) × (–) = – (+)  (–) = – (–) × (–) = + (–)  (–) = +

A simple rule for remembering the above is that if the signs are the same when multiplying or dividing, the answer will be positive and if the signs are different, the answer will be negative

Adding

Adding the same sign results in a sum of the same sign:

(+) + (+) = + and (–) + (–) = –

When adding numbers of different signs, follow this two-step process:

1 Subtract the absolute values of the numbers.

2 Keep the sign of the larger number.

Examples:

–2 + 3 =

1 Subtract the absolute values of the numbers:

3 – 2 = 1

2 The sign of the larger number (3) was originally

positive, so the answer is positive 1

8 + –11 =

1 Subtract the absolute values of the numbers:

11 – 8 = 3

2 The sign of the larger number (11) was originally

negative, so the answer is –3

Subtracting

When subtracting integers, change all subtraction to addition and change the sign of the number being sub-tracted to its opposite Then follow the rules for addition

Examples:

(+10) – (+12) = (+10) + (–12) = –2 (–5) – (–7) = (–5) + (+7) = +2

Decimals

The most important thing to remember about decimals

is that the first place value to the right begins with tenths The place values are as follows:

In expanded form, this number can also be expressed as

1,268.3457 = (1 × 1,000) + (2 × 100) + (6 × 10) + (8 × 1) + (3 × 1) + (4 × 01) + (5 × 001) + (7 × 0001)

1 T H O U S A N D S

2 H U N D R E D S

6 T E N S

8 O N E S

• D E C I M A L

3 T E N T H S

4 H U N D R E D T H S

5 T H O U S A N D T H S

7 T E N T H O U S A N D T H S

POINT

Comparing Decimals

Comparing decimals is actually quite simple Just line

up the decimal points and fill in any zeroes needed to

have an equal number of digits

Example:

Compare 5 and 005

Line up decimal points 500

Then ignore the decimal point and ask, which is bigger: 500 or 5?

500 is definitely bigger than 5, so 5 is larger than 005

Fractions

To do well when working with fractions, it is necessary

to understand some basic concepts Here are some

math rules for fractions using variables:

a

b×

d c= 

b

a

××d

c

b+ 

b c= a +

b c



a

b÷

d c= a

b×d

c= a

b

×

×

d c

 a

b+ 

d c= ad

b

+

d

bc



Multiplying Fractions

Multiplying fractions is one of the easiest operations to

perform To multiply fractions, simply multiply the

numerators and the denominators, writing each in the

respective place over or under the fraction bar

Example:

4

5 ×6

7 = 2 3

4 5



Dividing Fractions

Dividing fractions is the same thing as multiplying

fractions by their reciprocal To find the reciprocal of

any number, switch its numerator and denominator

For example, the reciprocals of the following numbers are:

1

3 = 3

1 = 3 x = 1

5 = 5

4  5 = 1

5 

When dividing fractions, simply multiply either fraction by the other’s reciprocal to get the answer

Example:

1 2

2 1

÷3

4 = 1 2

2 1

×4

3 = 4 6

8 3

= 1 2

6 1



Adding and Subtracting Fractions

■ To add or subtract fractions with like denomina-tors, just add or subtract the numerators and leave the denominator as it is For example,

1

7 + 5

7 = 6

7  and 5

8 – 2

8 = 3

8 

■ To add or subtract fractions with unlike

denomi-nators, you must find the least common denominator, or LCD.

For example, if given the denominators 8 and 12, 24 would be the LCD because 8 × 3 = 24 and 12 × 2 = 24

In other words, the LCD is the smallest number divis-ible by each of the denominators

Once you know the LCD, convert each fraction to its new form by multiplying both the numerator and denominator by the necessary number to get the LCD, and then add or subtract the new numerators

Example:

1

3 + 2

5 = 5 5

( (

1 3

) )

+ 3 3

( (

2 5

) )

=  1

5 5

+  1

6 5

= 1 1

1 5



Sets

Sets are collections of numbers and are usually based on certain criteria All the numbers within a set are called

the members of the set For example, the set of integers

looks like this:

{ –3, –2 , –1, 0, 1, 2, 3, }

The set of whole numbers looks like this:

{ 0, 1, 2, 3, }

When you find the elements that two (or more)

sets have in common, you are finding the intersection

of the sets The symbol for intersection is:∩

For example, the intersection of the integers and the whole numbers is the set of the whole numbers itself This is because the elements (numbers) that they have in common are {0, 1, 2, 3, } Consider the set

of positive even integers and the set of positive odd integers The positive even integers are:

{2, 4, 6, 8, }

The positive odd integers are:

{1, 3, 5, 7, }

If we were to combine the set of positive even numbers with the set of positive odd numbers, we

would have the union of the sets:

{1, 2, 3, 4, 5, }

The symbol for union is:∪

Mean, Median, and Mode

To find the average or mean of a set of numbers, add

all of the numbers together and divide by the quantity

of numbers in the set

Average = 

qu

n a

u n

m ti

b ty

er o

s f

e s

t et



Example:

Find the average of 9, 4, 7, 6, and 4

9 + 4 + 7 5

+ 6 + 4

= 3

5

0

= 6 (because there are 5 numbers in the set)

To find the median of a set of numbers, arrange the

numbers in ascending order and find the middle value

■ If the set contains an odd number of elements, then simply choose the middle value

Example:

Find the median of the number set: 1, 5, 3, 7, 2 First, arrange the set in ascending order: 1, 2, 3,

5, 7, and then, choose the middle value: 3 The answer is 3

■ If the set contains an even number of elements, simply average the two middle values

Example:

Find the median of the number set: 1, 5, 3, 7, 2, 8 First, arrange the set in ascending order: 1, 2, 3, 5,

7, 8, and then, choose the middle values 3 and 5 Find the average of the numbers 3 and 5:3 +25

= 4 The answer is 4

The mode of a set of numbers is the number that

occurs the greatest number of times

Example:

For the number set 1, 2, 5, 3, 4, 2, 3, 6, 3, 7, the number 3 is the mode because it occurs the most number of times

Percent

A percent is a measure of a part to a whole, with the whole being equal to 100

■ To change a decimal to a percentage, move the decimal point two units to the right and add a percentage symbol

Examples:

.45 = 45% 07 = 7% 9 = 90%

■ To change a percentage to a decimal, simply move the decimal point two places to the left and elimi-nate the percentage symbol

Examples:

64% = 64 87% = 87 7% = 07

■ To change a fraction to a percentage, first change the fraction to a decimal To do this, divide the numerator by the denominator Then change the decimal to a percentage

Examples:

4

5 = 80 = 80% 2

5 = 4 = 40% 1

8 = 125 = 12.5%