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** multiplying terms with the same base **
^{4 + }7^{6}
^{n}√x
^{n}√x * ^{n}√y =^{2}√10 * ^{2}√5

^{n}√xy^{2}√50

^{b}√x^{a =}

What are integers?

Whole Numbers, that are positive or negative (2 or -2)

Define a is divisible by b?

An integer A can be divided by b with out a remainder ie: result in a integer

Define Quotient

The result of two integers being divided ex: 8/2=4

Other forms of divisible?

divisior and factor ie: 8/2 the "2" is a divisor and "2" is a factor fo 8

IF a number is divisible by 2?

Number is even

number is divisible by 3

IF sum of integer's digits is divisible by 3

number is divisible by 4

If the integer is divisible by 2 TWICE or the last two digits are divisible by 4

number is divisible by 6

if the integer is divisible by both 2 and 3

number is divisible by 8

IF integer is divisible by 2 three times or if the last three digits are divisible by 8

number is divisible by 9

if sume of integers is divisible by 9

what is a factor of an integer?

is a positive integer that divides evenly into an integer example: 1,2,4,8 are all factors of 8. Factors since they divid into the integer are less than or equal to the integer

what is a multiple of an integer?

Is an integer formed by multiplying that integer by any integer. Ie: 8,16,24…are multiples of 8. b/c 8x1=8 and so on. B/c multiples multiply out from the integer so they are equal to or greater than the integer

12 is divisible by 3

12/3 remainder of 0 ; 3 divides 12; 3 is a divisor of 12, or 3 is a factor of 12; 3 goes into 12 evenly

12 items can be shared among 3 people so that each person has the same number of items.

12/3

12 is a multiple of 3

which means 12 is divisible by 3 and 12=3n?

What happens if you add or subtracts two multiples of an integer

you get a multiple ie: 7+14=21

What are the first 10 primes?

2, 3, 5, 7, 11, 13, 17, 19, 23, 29

If a is a factor of b and b is a factor of c?

Then a is a factor of c. any integer is divisibile by al of its factors and it's factors factors. IE: 72 is divisible by 8 and 8 is divisible by 2 and 4. Thus 72 is also divisible by 2 and 4. So 2 and 4 are factors of 72

When to use prime factoriziation ?

1. Determining whether one number is divisible y antehr number (2) Determining the greatest common factor of two numbers (3) reducing fractions (4) finding the least common multiple of two (or more) numbers (5) simplifying square roots (6) Determining the exponet on one side of an equation with integer constraints

what is the partial prime box of n?

n is divisible by a # of primes but stil don't what ADDITIONAL primes n has.

What is the difference btween a partial prime box and a full prime box?

partial prime box is for a varaible and full prime box is for a number

What is the Greatest Common Factor?

GCF is the largest divisior of two or more integers. THE GCF is the product of SHARED of primes. Take the lowest of the primes. Ie: 24 and 30: 24 (2^3 and 3^1) 30 (3^1 2^1 5^1). The common factors to the lowest power are 3^1 and 2^1. So the GCF is 6.

What is the Least Common Multiple?

The product of All the primes. So take the primes to the greatest power: Ie: 24 and 30: 24 (2^3 and 3^1) 30 (3^1 2^1 5^1). The LMC, primes to the highest power are 3^1 and 2^3 and 5^1. So the LMC 120

IF a is divided by 7 or by 18, an integer results. Is a/42 an integer?

a has the following primes 7, 3,3,2 . a/42. 42 is a multiple of 7 & 18. 42 has primes 7,2,3 thus must be divisible into a b/c any product that can be construcuted by the factors of 7 and 18 must also be a factor of a.

Add or subtract 2 odds? Add or subtract two evens?

The result is EVEN

Add or subtract an odd with an even ?

The result is ODD

Multiply any integer with an even

Result is even

odd x odd

result is odd

odd divided by odd

never even, but can be ODD or a NON INTEGER

odd divided by even

can NEVER even and can NEVER be ODD b/c the odd number can never be divisible by the factor of 2 concealed with in the even number

even divided by odd

can be even, can never be ODD

even divided by an even

can be even or odd (10/2 =5 or 8/2=4)

Are all primes ODD?

Yes except for 2 which is even

When you add two primes?

You are adding two odds thus the result is even. Unless one of them is even in which it is odd+even=odd

Define absolute value?

how far is the number from the 0 on the number line? -5 is 5 units from 0 as is 5 from 0.

If x=-y

IF two numbers are opposties fo each other, then they have the same absolute value, and 0 is halfway between them. We cannot tell which variable is positive without more information

PEMDAS

Prenteseses , Exponenens,division/ multiplication, addtion/subtraction

If there are 5 negative numbers in Set S is the product neagative?

There are EXZATLY -5 numbers, the rest of the numbers in the set are even or 0. The product can be negative or 0. You don't if any of the other numbers in the set are 0. In which case the product will be 0. [answer is insufficient]

All of elements in SET S are negative

Then if there are 2 negative numbers then answer is positive, if there are 3 then it is negative. Issue you don't know the number of elements in the set. [insufficient]

if ab>0, which of the following must be negative?

This means that the signs of a and b must both be the same (both positive or both negative). The answer is -(a/b) wil always be negative. To solve set up a table to track the negatives and positives

what are evenly spaced sets?

sequences of numbers whose values go up or down by the same amt {increment}

What are consequtive integers?

all values in the set are multiples fo the increment

conequetive integers

all the values in the set increase by 1

All evenly spaced sets are fully defined if the following 3 parapeters are know:

(1) The smallest (first number) or largest (last) number in the set (2) The increment (always 1 for consecutive integers (3) The number of items in the set

What are the properties of evenly spaced sets?

1.arithmetic mean (average) and medican are equal to each other 2. The mean and median of the set are equal to the average of the FIRST and LAST terms 3. sum of the elements equals the arthmetic mean number in the set times the number of times in the set

counting integers formula

IF you are counting intengers wich are inclusive. You must add +1 to include the first exterem which is being subtracted away. (last-first)+1

How many integers are there from 14 to 765, inclusive?

765-14 = 751+1 = 752

counting consequtive multiples in a set

(last-first)/(increment) +1

How many multiples of 7 are there between 100 and 150?

Counting numbers (147-105)/7= 6+1=7 TERMS. To identify the first term divide 7 into a hundred to see what the next multiple of 7 would be.

Find the sum of all the integers inclusive

(1) find number of items in the set (2) middle number (median) of the set (3) muliple #s and median

Average of of an even set

Average of of an even set is not an integer. b/c consequtive integers alternative between EVEN AND OOD. The middle number of an even and odd is not an integer

the sum of k consecutive integers is divisible by k

this is the defintion of an average. This means the set of k consequtives integers is an odd number set. (sum of k integers)/k = (average of k integers) = integer

average of odd number of consequtive integers?

the result is always an integer

the product of K consecutive integers is dvisible by?

Always divisible by k factorial (K!) ie: The product of 3 consequitve integers will always be divisible by 3! 3x2x1=6. B/c there is always a multiple of 3, 2 in any set of 3 consequitve integers. 2x3x4=24

For any set of consecutive integers with an ODD number of numbers, the sum of all the integers is ALWAYS?

The sum of all fo the integers is always a multiple of the number of items. b/c the sum equals the average times the number of times. 2+3+4=9 and sum= (average is )3 x (# of times is 3).

For a set of consecutive integers with an EVEN number of items is the sum a mulltiple of the number of items?

NO. B/c The sum equal the average times the number of item. For a n even number of evenly spaced integers the average is never an integer

How do you express: There is a set of 7 consecutive integers

n, n+1, n+2, n+3, n+4, n+5, n+6

If x is an even integer, is x(x+1)(x+2) divisible by 4?

Yes. (x will have a2), and x+1 will be some odd integer and (x+2 will have a 2). Method: set up three connect prime boxes.

Exponents 5^{2}.

5 is the base and 2 is exponent

exponent 2 and 3

they are called squares and cubes

greater the exponent

the faster the rate of increase

Why do you have to watch out for an even exponent

the base can be negative or positive and the result will always be positve

Why are odd exponentes "harmless"?

The base keeps the original sign under an odd exponent

An exponential expression with a base of 0

always yeilds a 0 regardless of the exponent

An exponential expression with a base of 1 always yeilds?

always yeilds 1 regardless of the exponent

An exponential expression with a base of -1

yeilds -1 when the exponent is odd and yeilds 1 when the expoent is even

If you are told x^{6}=x^{8}=x^{10}

x can be 0,1 or -1

what happens to fractional bases as the exponents increase?

The bases decrease. (3/4) vs (3/4)^{2} = 9/16

The base of an exponential expression is a product

(2 * 5)^{3}

a^{x} * b^{x} =

Multiply base together and raise it to the exponent. Or distribute the exponent to each number in the base and then multiply. IE: (2* 5)^{3}=10^{3} or 2^{3*}5^^{3}=8 * 125=1000

a^{x} * b^{x} = ( ab)^{x}

What can you do with bases that are being summed? (2+5)^{3}

You cannot distibute the exponent to each of the bases with in the sum. You must add and then carry out the exponent. (2+5)^{3}=7^{3}=343

{ 3^{3} * 3^{2}}=

{x^{a} * x^{b}} =

Add the exponents

{ 3^{3} * 3^{2}} = 3^{(5)}

{x^{a} * x^{b}}^{ = }x^{a+b}

dividing two terms with the same base

(3^{3} ÷ 3^{2})=

x^{a} ÷ x^{b} =

Combine exponnts by subtraction

( 3^{3} ÷ 3^{2}) = 3^{(3-2) }= 3^{1}

x^{a} ÷ x^{b} = x ^{(a-b)}

dividing two terms with the same exponent (a ÷ b)^{x} =

a^{x}/a^{b}

Raising a power to a power (3^{2})^{4}

(a^{x})^{y} =

combine exponents by multiplciation (3^{2})^{4}= 3^{12}

(a^{xy}) or (a^{y})^{x}

Negative exponents

(5 ^{-1})

x^{-a}

the base is reciprocal (5^{-1}) = (1/5)

=1/x^{a}

any non zero base raised to the 0 power yeilds? (-1/2)^{0} or 4^{0}

1.0

Fractional exponents (25 ^{3/2})

x^{(a/b)}

numerator tells us the power to raise the base to and the dnominaotr tells us which root to take. 25^{3} is (5^{2})^{3} so root of (5^{2})^{3 }is √ 5^{6} = 5^{3}. You can also right (5^{2}) ^{3/2} =5^{3}^{b}√x^{a =} (^{b}√x)^{a}

a^{x}+a^{x}+a^{x} =?

3a^{x}

3^{x}+3^{x}+3^{x} =?

3 * 3x = 3x+1

When can you simplify exponential expressions?

When they are linked by multiplication or division.__ Also they must have either a common base or common exponent. __

Can the following be simplified?

76^{5 -} 6^{3}

3^{4 + }12^{4}

12^{7 - }3^{7}

You CANNOT simplify those linked by addition or subtraction__.__They can be factored ie:

7^{4}(1+7^{2})

6^{3}(6^{2}-1)

12^{4} = (2*2*3)^{4 = }same bases are 3^{4 }so 3^{4}(1+4^{4})

3^{7}( 4^{7}-1)

Even roots have only what kind of value?

Even roots only have a positive value. √4 = 2 not ±2. However when you decided to unsquare an equation with even exonents you need to consider both positive and negativ solutions. EX: x^{2}= 4 (x=2 and x=-2)

When can a root have a negative value?

- it is an odd root
^{ 3}√**AND** - the base of the root is negative
^{3}√-27 = -3

Define the parts of the exponential fraction

x^{a/b}

^{b}√x^{a}

Can you combine or sperate roots in addition or subtraction?

NO √x+y cannot be simplified.

When can you simplify roots?

Only by multiplication or division

— or √10

—

^{n}√y √5

x^{n}√— = √(10/5) = √2

y

49 ^{-1/2}

25^{3/2}

x^{a/b}

√1/49 = 1/7^{}

√25^{3 = }5^{3 = }125

What can be done with imperfect squares?

√52

you can simplify them

√2 * 2 * 13 = 2√13

1^{2}

1

1.4^{2}

≈ 2

1.7^{2}

≈3

2.25^{2}

≈5

2^{2}

4

3^{2}

9

4^{2}

16

5^{2}

25

6^{2}

36

7^{2}

49

8^{2}

64

9^{2}

81

10^{2}

100

11^{2}

121

12^{2}

144

13^{2}

169

14^{2}

196

15^{2}

225

16^{2}

256

20^{2}

400

25^{2}

625

30^{2}

900

√1

1

√2

≈1.4

√3

1.7

√5

1.25

√4

2

√169

13

√196

14

√225

15

√256

16

√400

20

√625

25

√900

30

2^{3}

8

3^{3}

27

4^{3}

64

5^{3}

125

s_ _ _ .... r

Total numbers from s to r?

s - r + 1

s_ _ _ .... r

Total numbers from s to r, not including s and r?

= s - r + 1 -2

= s - r -1

About this deck

Author: Cindy N.

Created: 2014-05-12

Updated: 2014-05-14

Size: 130 flashcards

Views: 4

Created: 2014-05-12

Updated: 2014-05-14

Size: 130 flashcards

Views: 4

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