Chapter 5-2 FRACTIONS = DECIMALS
How to change a fraction to a decimal
1. Divide (ALWAYS WORKS!)
EXAMPLE: 3/4 =
3 divided by 4 =
0.75
If the quotient starts repeating, then put a bar over the number(s) that repeat.
OR
2. Use equivalent fractions (SOMETIMES WORKS!)
Works if the denominator can be easily made into a power of 10
SAME EXAMPLE: 3/4
but this time you will multiply by 25/25 to get
75/100 = 0.75
3. MEMORY! Some equivalencies you should just know!
EXAMPLE: 1/2 = 0.5
IF IT'S A MIXED NUMBER, JUST ADD THE WHOLE NUMBER AT THE END!
EXAMPLE: 8 3/4
For the fraction: 3 divided by 4 = .75
Add the whole number:
8.75
IF THE MIXED NUMBER OR FRACTION IS NEGATIVE, SO IS THE DECIMAL!
Changing decimals to fractions.
CHANGING TERMINATING DECIMALS TO FRACTIONS:
EASY!!!
I learned this catchy phrase READ IT WRITE IT REDUCE!!
But we no longer say "reduce". We now say "simplify"
EXAMPLE:
Change .24 to a fraction
1) READ IT: 24 hundredths
2) WRITE IT: 24/100
3) SIMPLIFY: 24/100 = 6/25
EXAMPLE with whole number:
Change 7.24 to a fraction
The 7 is the whole number in the mixed number so you just put the 7 at the end
1) READ IT: 24 hundredths
2) WRITE IT: 24/100
3) SIMPLIFY: 24/100 = 6/25
4) (Now is the time to put the whole number back!!) 7 6/25
HOW TO CHANGE REPEATING DECIMALS TO FRACTIONS:
Repeating decimals (we'll use algebra!)
This involves algebra and takes some work, so MEMORIZE THE FOLLOWING:
1/3 = .333 . . . and 2/3 = .666. . .
Also showed the 1/9 family pattern which is the numerator with a bar
Another great time-saving pattern - 1/11 family: Multiply the numerator by 9 and put a bar over it
To get an exact answer when doing math operations with repeating decimals,
make all repeating decimals into their fraction equivalents and do the operations with fractions!
To change repeating decimals to fractions, follow these steps:
Let n = the repeating decimal
Multiply both sides by a power of 10 equal to the number of places that repeat under the bar
Subtract n on the left side and the repeating decimal equal to n on the right side
Solve as a one-step equation
Multiply both numerator and denominator by a power of 10 if necessary
to get the decimal out of the numerator.
Simplify
EXAMPLE: Restate .41666 . . . into a fraction
The repeating portion is .6 or one place so multiply both sides by 10
n = .41666 . . .
10n = 4.1666 . . .
- n = -.4166 . . .
9n = 3.75 so divide both sides by 9 to get
9n/9 = 3.75/9
n = 375/900
n = 5/12
Wednesday, January 7, 2009
MULTIPLICATION OF MONOMIALS AND BINOMIALS 5-9
Multiplying a monomial by a polynomial is just the Distributive Property
You'll need to remember your power rules because you'll be MULTIPLYING SAME BASES
and then combining only LIKE TERMS.
EXAMPLE:
3x3 ( 2x2 - 3x + 10)
3x3(2x2) + 3x3(- 3x) + 3x3(10)
6x5 - 9 x5 + 30x3
(no like terms to combine in this one!)
When you multiply a binomial by another binomial, I used to think of it as DOUBLE Distributive Property...See if you can figure out why!
Multiplying a binomial by another binomial:
FOILing binomials:
This is a memory device so you won't forget to multiply any of the factors!
First - first term in each binomial (terms on the left)
Outside - two outside terms in each binomial ( the first one in the
left parentheses and the second one in the right parentheses)
Inside - two inside terms (the terms right next to each other in the
different parentheses - the second one in the first parentheses
and the first one in the second parentheses)
Last - second term in each binomial (terms on the right)
I know this may seem overwhelming when you first see it,
but after you practice it, it does make sense.
It helps a lot of students to not forget any of the 4 multiplications!
EXAMPLE: (5x + 6)(3x - 7)
F O I L
FIRST OUTSIDE INSIDE LAST
= (5x)(3x) + (5x)(-7) + (6)(3x) + (6)(-7)
Simplify: 15x2 + (-35x) + 18x - 42
Combine like terms: 15x2 + -17x - 42
By the way, you can also do this in different orders --- as long as you multiply each term in one parentheses by each term in the other parentheses.
There are other ways to do the same problem:
Showed you the "box" method that allows you to use any order and protects you from ever missing one of the four multiplications. ( I can't get a table embedded here so make sure to look at your notes about the box method-- or email me.
5x +6
3x 15x2 18x
-7 -35x -42
15x2+ 18 x – 35x -42 =
15x2 – 17x – 42
Also showed you "unibrow" method where you use arrows to show which terms you are multiplying. THIS IS HOW I LEARNED AND SEE IF YOU CAN FIGURE OUT WHY I USED TO THINK OF IT AS DOUBLE DISTRIBUTIVE PROPERTY!
I can't really show this on this website so watch carefully in class!
Multiplying a monomial by a polynomial is just the Distributive Property
You'll need to remember your power rules because you'll be MULTIPLYING SAME BASES
and then combining only LIKE TERMS.
EXAMPLE:
3x3 ( 2x2 - 3x + 10)
3x3(2x2) + 3x3(- 3x) + 3x3(10)
6x5 - 9 x5 + 30x3
(no like terms to combine in this one!)
When you multiply a binomial by another binomial, I used to think of it as DOUBLE Distributive Property...See if you can figure out why!
Multiplying a binomial by another binomial:
FOILing binomials:
This is a memory device so you won't forget to multiply any of the factors!
First - first term in each binomial (terms on the left)
Outside - two outside terms in each binomial ( the first one in the
left parentheses and the second one in the right parentheses)
Inside - two inside terms (the terms right next to each other in the
different parentheses - the second one in the first parentheses
and the first one in the second parentheses)
Last - second term in each binomial (terms on the right)
I know this may seem overwhelming when you first see it,
but after you practice it, it does make sense.
It helps a lot of students to not forget any of the 4 multiplications!
EXAMPLE: (5x + 6)(3x - 7)
F O I L
FIRST OUTSIDE INSIDE LAST
= (5x)(3x) + (5x)(-7) + (6)(3x) + (6)(-7)
Simplify: 15x2 + (-35x) + 18x - 42
Combine like terms: 15x2 + -17x - 42
By the way, you can also do this in different orders --- as long as you multiply each term in one parentheses by each term in the other parentheses.
There are other ways to do the same problem:
Showed you the "box" method that allows you to use any order and protects you from ever missing one of the four multiplications. ( I can't get a table embedded here so make sure to look at your notes about the box method-- or email me.
5x +6
3x 15x2 18x
-7 -35x -42
15x2+ 18 x – 35x -42 =
15x2 – 17x – 42
Also showed you "unibrow" method where you use arrows to show which terms you are multiplying. THIS IS HOW I LEARNED AND SEE IF YOU CAN FIGURE OUT WHY I USED TO THINK OF IT AS DOUBLE DISTRIBUTIVE PROPERTY!
I can't really show this on this website so watch carefully in class!
Math 6 Honors Periods 6 & 7 (Tuesday)
Adding Integers 11-2
Rules: The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
So- if the two numbers have the same sign, use their sign and just add the numbers.
-15 + -13 = - 28
-10 + -4 = -14
Rules: The sum of a positive integer and a negative integer is :
POSITIVE… IF the positive number has a greater absolute value
NEGATIVE… IF the negative number has a greater absolute value
ZERO… IF both numbers have the same absolute value
Think of a game between two teams-
The POSITIVE TEAM vs. The NEGATIVE TEAM.
30 + -16 … ask yourself the all important question…
“WHO WINS? in this case the positive and then ask
“BY HOW MUCH?” take the difference 14
14 + - 52…
“WHO WINS?” the negative…
“BY HOW MUCH?”
38 so 14 + (-52) = -38
Although Aunt Sally says you must do ( ) first, when you are ONLY adding you can use the Commutative and Associative Properties to aid you!!
(-2 + 3) + - 6 you can work this 2 ways
(-2 + 3) + - 6 = 1 + -6 = -5 or
using all the properties that work for whole numbers
Commutative and Associative Properties of Addition
can change expression to (-2 + -6) + 3 or -8 + 3 = -5 you still arrive at the same solution.
You want to use these properties when you are adding more than 2 integers.
First look for zero pairs—you can cross them out right away!!
3 + (-3) = 0
-9 + 9 = 0
Then you can use C(+) to move the integers around to make it easier to add them together rather than adding them in the original order. In addition, you can use A(+) to group your positive and negative numbers in ways that make it easier to add as well.
One surefire way is to add all the positives up… and then add all the negatives up. At this point ask yourself that all important question…
WHO WINS? … use the winner’s sign..
and then ask yourself..
BY HOW MUCH?
example:
-4 + 27 +(-6) + 5 + (-4) + (6) + (-27) + 13
Taking a good scan of the numbers, do you see any zero pairs?
YES—so cross them out and you are left with
-4 + 5 + (-4) + 13
add your positives 5 + 13 = 18
add your negatives and use their sign – 4 + -4 = -8
Okay, Who wins? the positive
By how much? 10
so
-4 + 27 +(-6) + 5 + (-4) + (6) + (-27) + 13 = 10
Rules: The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.
So- if the two numbers have the same sign, use their sign and just add the numbers.
-15 + -13 = - 28
-10 + -4 = -14
Rules: The sum of a positive integer and a negative integer is :
POSITIVE… IF the positive number has a greater absolute value
NEGATIVE… IF the negative number has a greater absolute value
ZERO… IF both numbers have the same absolute value
Think of a game between two teams-
The POSITIVE TEAM vs. The NEGATIVE TEAM.
30 + -16 … ask yourself the all important question…
“WHO WINS? in this case the positive and then ask
“BY HOW MUCH?” take the difference 14
14 + - 52…
“WHO WINS?” the negative…
“BY HOW MUCH?”
38 so 14 + (-52) = -38
Although Aunt Sally says you must do ( ) first, when you are ONLY adding you can use the Commutative and Associative Properties to aid you!!
(-2 + 3) + - 6 you can work this 2 ways
(-2 + 3) + - 6 = 1 + -6 = -5 or
using all the properties that work for whole numbers
Commutative and Associative Properties of Addition
can change expression to (-2 + -6) + 3 or -8 + 3 = -5 you still arrive at the same solution.
You want to use these properties when you are adding more than 2 integers.
First look for zero pairs—you can cross them out right away!!
3 + (-3) = 0
-9 + 9 = 0
Then you can use C(+) to move the integers around to make it easier to add them together rather than adding them in the original order. In addition, you can use A(+) to group your positive and negative numbers in ways that make it easier to add as well.
One surefire way is to add all the positives up… and then add all the negatives up. At this point ask yourself that all important question…
WHO WINS? … use the winner’s sign..
and then ask yourself..
BY HOW MUCH?
example:
-4 + 27 +(-6) + 5 + (-4) + (6) + (-27) + 13
Taking a good scan of the numbers, do you see any zero pairs?
YES—so cross them out and you are left with
-4 + 5 + (-4) + 13
add your positives 5 + 13 = 18
add your negatives and use their sign – 4 + -4 = -8
Okay, Who wins? the positive
By how much? 10
so
-4 + 27 +(-6) + 5 + (-4) + (6) + (-27) + 13 = 10
Sunday, January 4, 2009
Math 6 H Periods 1, 6 & 7 (Review)
Negative Numbers 11-1
On a horizontal number line we use negative numbers for the coordinates of points to the left of zero. We denote the number called ‘negative four’ by the symbol -4. The symbol -4 is normally read ‘ negative 4’ but we can also say ‘ the opposite of 4.’
The graphs of 4 and -4 are the same distance from 0—but in opposite directions. Thus they are opposites. -4 is the opposite of 4.
The opposite of 0 is 0
Absolute Value is a distance concept. Absolute value is the distance of a number from 0 on a number line. The absolute value of a number can NEVER be negative!!
Counting (also known as Natural) numbers: 1, 2, 3, 4, ….
Whole numbers 0, 1, 2, 3, 4….
Integers are natural numbers and their opposites AND zero
…-4, -3, -2, -1, 0, 1, 2, 3, 4….
The opposite of 0 is 0.
The integer 0 is neither positive nor negative.
The farther we go to the right on a number line--- the bigger the number. We can compare two integers by looking at their position on a number line.
if x < 0 what do we know? x is negative number
if x > 0, what do we know? x is a positive number
We have been practicing representing integers by their graphs, that is, by points on a number line. Make sure that your number line includes arrows at both ends and a line indicating where zero falls on your number line. The graph of a number MUST have a closed dot right on the number line at that specific number. Please see our testbook page 366 for an accurate example.
On a horizontal number line we use negative numbers for the coordinates of points to the left of zero. We denote the number called ‘negative four’ by the symbol -4. The symbol -4 is normally read ‘ negative 4’ but we can also say ‘ the opposite of 4.’
The graphs of 4 and -4 are the same distance from 0—but in opposite directions. Thus they are opposites. -4 is the opposite of 4.
The opposite of 0 is 0
Absolute Value is a distance concept. Absolute value is the distance of a number from 0 on a number line. The absolute value of a number can NEVER be negative!!
Counting (also known as Natural) numbers: 1, 2, 3, 4, ….
Whole numbers 0, 1, 2, 3, 4….
Integers are natural numbers and their opposites AND zero
…-4, -3, -2, -1, 0, 1, 2, 3, 4….
The opposite of 0 is 0.
The integer 0 is neither positive nor negative.
The farther we go to the right on a number line--- the bigger the number. We can compare two integers by looking at their position on a number line.
if x < 0 what do we know? x is negative number
if x > 0, what do we know? x is a positive number
We have been practicing representing integers by their graphs, that is, by points on a number line. Make sure that your number line includes arrows at both ends and a line indicating where zero falls on your number line. The graph of a number MUST have a closed dot right on the number line at that specific number. Please see our testbook page 366 for an accurate example.
Algebra Period 3 (Review)
REVIEW:WEEK BEFORE WINTER BREAK
Sections 5-5 to 5-8 Polynomials
Polynomial = sum of monomials
Monomials must have variables with WHOLE NUMBER powers
(constants have whole number powers because you can say it has a variable to the zero)
(no variables in the denominator and no roots of numbers)
1 Term = monomial
2 terms = binomial
3 terms = trinomial
Terms are separated by addition
(and subtraction...although we never subtract...we add the opposite.)
Coefficient = Number attached to variable (can be a fraction!)
The sign of the coefficient should be looked at AFTER you change any subtractions to addition
For example: 3x2 - 10x
The coefficients are 3 and -10
Also, if you have y/6, you really have 1y/6 (ID prop of multiplication)
so the coefficient of y/6 is 1/6 and can be written as (1/6) y
if you have -y/6, you really have -1y/6 (ID prop of multiplication)
so the coefficient of -y/6 is -1/6 and can be written as (-1/6) y
Constant = the number that is not attached to any variable
Degree of a term = sum of the exponents of all its VARIABLES
Degree of a polynomial = HIGHEST degree of any of its terms
Leading term = term with the HIGHEST degree
Leading coefficient = the coefficient of the LEADING TERM
Sections 5-6
DESCENDING ORDER - Write the variables with the highest power first
ASCENDING ORDER - Write the variable with the lowest power first
(this order is actually never used in practice!)
EVALUATING A POLYNOMIAL - WE'VE BEEN DOING THIS ALL YEAR! Plug it in, plug it in, plug it in! Then use Aunt Sally!
Remember to ALWAYS put the number you substitute in parentheses!!!
Sections 5-7 ADDING POLYNOMIALS
This is nothing more than combining LIKE TERMS
LIKE TERMS = same variable AND same power
You can either do this using 3 different strategies:
1. Simply do it in your head, but keep track by crossing out the terms as you use them.
2. Rewrite putting the like terms together (commutative and associative property)
3. Rewrite in COLUMN form, putting like terms on top of each other like you do when adding a column of numbers.
EXAMPLE OF COLUMN FORM:
(5x4 - 3x2 - (-4x) + 3) + (-10x4 + 3x3- 3x2 - x + 3)
Rewrite in column form, lining up like terms:
Section 5-8: SUBTRACTING POLYNOMIALS
You can use the ADDITIVE INVERSE PROPERTY with polynomials!
Subtracting is simply adding the opposite so.............
DISTRIBUTE THE NEGATIVE SIGN TO EACH TERM!!
(Change all the signs of the second polynomial!)
After you change all the signs, use one of your ADDING POLYNOMIAL strategies!
(see the 3 strategies listed above under Chapter 5-7)
EXAMPLE OF COLUMN FORM:
(5x4 - 3x2 - (-4x) + 3) - (-10x4 + 3x3- 3x2 - x + 3)
Rewrite in column form, lining up like terms:
5x4 - 3x2 - (-4x) + 3
- ( -10x4 + 3x3- 3x2 - x + 3)
-----------------------------------
For the sake of showing you here, I have added ZERO Terms to line up columns
+ 5x4 + 0x3 - 3x2 -(-4x) + 3
-(-10x4 +3x3- 3x2 - x + 3)
-----------------------------------
DISTRIBUTE THE NEGATIVE, THEN ADD:
5x4 + 0x3 - 3x2 - (-4x) + 3
+10x4 -3x3 +3x2 + x - 3
-----------------------------------
15x4 - 3x3 + 5 x
Sections 5-5 to 5-8 Polynomials
Polynomial = sum of monomials
Monomials must have variables with WHOLE NUMBER powers
(constants have whole number powers because you can say it has a variable to the zero)
(no variables in the denominator and no roots of numbers)
1 Term = monomial
2 terms = binomial
3 terms = trinomial
Terms are separated by addition
(and subtraction...although we never subtract...we add the opposite.)
Coefficient = Number attached to variable (can be a fraction!)
The sign of the coefficient should be looked at AFTER you change any subtractions to addition
For example: 3x2 - 10x
The coefficients are 3 and -10
Also, if you have y/6, you really have 1y/6 (ID prop of multiplication)
so the coefficient of y/6 is 1/6 and can be written as (1/6) y
if you have -y/6, you really have -1y/6 (ID prop of multiplication)
so the coefficient of -y/6 is -1/6 and can be written as (-1/6) y
Constant = the number that is not attached to any variable
Degree of a term = sum of the exponents of all its VARIABLES
Degree of a polynomial = HIGHEST degree of any of its terms
Leading term = term with the HIGHEST degree
Leading coefficient = the coefficient of the LEADING TERM
Sections 5-6
DESCENDING ORDER - Write the variables with the highest power first
ASCENDING ORDER - Write the variable with the lowest power first
(this order is actually never used in practice!)
EVALUATING A POLYNOMIAL - WE'VE BEEN DOING THIS ALL YEAR! Plug it in, plug it in, plug it in! Then use Aunt Sally!
Remember to ALWAYS put the number you substitute in parentheses!!!
Sections 5-7 ADDING POLYNOMIALS
This is nothing more than combining LIKE TERMS
LIKE TERMS = same variable AND same power
You can either do this using 3 different strategies:
1. Simply do it in your head, but keep track by crossing out the terms as you use them.
2. Rewrite putting the like terms together (commutative and associative property)
3. Rewrite in COLUMN form, putting like terms on top of each other like you do when adding a column of numbers.
EXAMPLE OF COLUMN FORM:
(5x4 - 3x2 - (-4x) + 3) + (-10x4 + 3x3- 3x2 - x + 3)
Rewrite in column form, lining up like terms:
Section 5-8: SUBTRACTING POLYNOMIALS
You can use the ADDITIVE INVERSE PROPERTY with polynomials!
Subtracting is simply adding the opposite so.............
DISTRIBUTE THE NEGATIVE SIGN TO EACH TERM!!
(Change all the signs of the second polynomial!)
After you change all the signs, use one of your ADDING POLYNOMIAL strategies!
(see the 3 strategies listed above under Chapter 5-7)
EXAMPLE OF COLUMN FORM:
(5x4 - 3x2 - (-4x) + 3) - (-10x4 + 3x3- 3x2 - x + 3)
Rewrite in column form, lining up like terms:
5x4 - 3x2 - (-4x) + 3
- ( -10x4 + 3x3- 3x2 - x + 3)
-----------------------------------
For the sake of showing you here, I have added ZERO Terms to line up columns
+ 5x4 + 0x3 - 3x2 -(-4x) + 3
-(-10x4 +3x3- 3x2 - x + 3)
-----------------------------------
DISTRIBUTE THE NEGATIVE, THEN ADD:
5x4 + 0x3 - 3x2 - (-4x) + 3
+10x4 -3x3 +3x2 + x - 3
-----------------------------------
15x4 - 3x3 + 5 x
Algebra Period 3 (Review)
REVIEW:
EXPONENTS SECTIONS 5-1 TO 5-3
SCIENTIFIC NOTATION SECTION 5-4
Review the odd/even rule
IF THERE IS A NEGATIVE INSIDE PARENTHESES:
Odd number of negative signs or odd power = negative
Even number of negative signs or even power = positive
EXAMPLES:
(-2)5 = -32
(-2)4 = +16
IF THERE IS A NEGATIVE BUT NO PARENTHESES:
ALWAYS NEGATIVE!!!!
-25 = -32
-24 = -16
MULTIPLYING Powers with LIKE BASES:
Simply ADD THE POWERS
m5m3 = m8
You can check this by EXPANDING:
(mmmmm)(mmm) = m8
DIVIDING Powers with LIKE BASES:
Simply SUBTRACT the POWERS
m8 / m5= m3
Again, you can check this by EXPANDING:
mmmmmmmm/mmmmm
cancel out
ZERO POWERS:
Anything to the zero power = 1
(except zero to the zero power is undefined)
Proof of this was given in class:
1 = mmmmmmmm/mmmmmmmm = m8 /m8 = m8-8 = m0
(by power rules for division)
By the transitive property of equality : 1 = m0
NEGATIVE POWERS = FRACTIONS
They're in the wrong place in the fraction!
NEGATIVE POWERS ARE NOT NEGATIVE NUMBERS!
THEY HAPPEN WHEN THERE IS A DIVISION OF LIKE BASES WHERE THE POWER ON THE TOP IS SMALLER THAN THE POWER ON THE BOTTOM!
WHEN YOU USE THE POWER RULES, YOU WILL SUBTRACT A BIGGER NUMBER FROM A SMALLER NUMBER AND THAT WILL CREATE A NEGATIVE POWER!
EXAMPLE:
m3 / m5 = m-2
m3 / m5 = mmm/mmmmm = 1/mm
Again, by transitive property of equality:
m3 / m5 = m-2 = 1/m2
EXPRESS NEGATIVE POWERS WITHOUT EXPONENTS:
1) MOVE TO DENOMINATOR
2) EXPAND THE POWER
EXAMPLE:
(-2)-5 = 1/(-2)5 = 1/-32 OR -1/32
RESTATE A FRACTION INTO A NEGATIVE POWER:
1) Restate the denominator into a power
2) Move to the numerator by turning the power negative
EXAMPLE:
1/32
1/(2)5
(2)-5
More with Exponents 5-2 &
Multiplying and Dividing Monomials 5-3
A monomial is an expression that is either a numeral, a variable, or a product of numerals and variables with whole number exponents.
POWER TO ANOTHER POWER
MULTIPLY the POWERS
(m5)3 = m15
To check, EXPAND it out:
(m5)(m5)(m5) = m15
PRODUCT TO A POWER
DISTRIBUTE the power to EACH FACTOR
(m5n4)3 = m15n12
RAISING A QUOTIENT TO A POWER:
DISTRIBUTE THE POWER to the numerator and the denominator
(m2/n6)3 = (m2)3/(n6)3 = m6/n18
Scientific Notation 5-4
You've had this since 6th grade!
You restate very big or very small numbers using powers of 10 in exponential form
Move the decimal so the number fits in this range: less than 10 and greater than or equal to 1 That is, 1 ≤ n < 10
Count the number of places you moved the decimal and make that your exponent
Very big numbers - exponent is positive
Very small numbers (decimals) - exponent is negative (just like a fraction!)
Remember that STANDARD notation is what you expect (the normal number)
When you multiply or divide scientific notations, use the power rules!
Just be careful that is your answer does not fit the scientific notation range, that you restate it.
TRY THIS LINK THAT TAKES YOU FROM HUGE POWERS TO
LITTLE TINY POWERS (NEGATIVE POWERS OR DECIMALS)
Try this link to practice scientific notation!
Try this link to practice multiplication of scientific notation!
Try this link to practice division!
Try this link to practice harder problems!
EXPONENTS SECTIONS 5-1 TO 5-3
SCIENTIFIC NOTATION SECTION 5-4
Review the odd/even rule
IF THERE IS A NEGATIVE INSIDE PARENTHESES:
Odd number of negative signs or odd power = negative
Even number of negative signs or even power = positive
EXAMPLES:
(-2)5 = -32
(-2)4 = +16
IF THERE IS A NEGATIVE BUT NO PARENTHESES:
ALWAYS NEGATIVE!!!!
-25 = -32
-24 = -16
MULTIPLYING Powers with LIKE BASES:
Simply ADD THE POWERS
m5m3 = m8
You can check this by EXPANDING:
(mmmmm)(mmm) = m8
DIVIDING Powers with LIKE BASES:
Simply SUBTRACT the POWERS
m8 / m5= m3
Again, you can check this by EXPANDING:
mmmmmmmm/mmmmm
cancel out
ZERO POWERS:
Anything to the zero power = 1
(except zero to the zero power is undefined)
Proof of this was given in class:
1 = mmmmmmmm/mmmmmmmm = m8 /m8 = m8-8 = m0
(by power rules for division)
By the transitive property of equality : 1 = m0
NEGATIVE POWERS = FRACTIONS
They're in the wrong place in the fraction!
NEGATIVE POWERS ARE NOT NEGATIVE NUMBERS!
THEY HAPPEN WHEN THERE IS A DIVISION OF LIKE BASES WHERE THE POWER ON THE TOP IS SMALLER THAN THE POWER ON THE BOTTOM!
WHEN YOU USE THE POWER RULES, YOU WILL SUBTRACT A BIGGER NUMBER FROM A SMALLER NUMBER AND THAT WILL CREATE A NEGATIVE POWER!
EXAMPLE:
m3 / m5 = m-2
m3 / m5 = mmm/mmmmm = 1/mm
Again, by transitive property of equality:
m3 / m5 = m-2 = 1/m2
EXPRESS NEGATIVE POWERS WITHOUT EXPONENTS:
1) MOVE TO DENOMINATOR
2) EXPAND THE POWER
EXAMPLE:
(-2)-5 = 1/(-2)5 = 1/-32 OR -1/32
RESTATE A FRACTION INTO A NEGATIVE POWER:
1) Restate the denominator into a power
2) Move to the numerator by turning the power negative
EXAMPLE:
1/32
1/(2)5
(2)-5
More with Exponents 5-2 &
Multiplying and Dividing Monomials 5-3
A monomial is an expression that is either a numeral, a variable, or a product of numerals and variables with whole number exponents.
POWER TO ANOTHER POWER
MULTIPLY the POWERS
(m5)3 = m15
To check, EXPAND it out:
(m5)(m5)(m5) = m15
PRODUCT TO A POWER
DISTRIBUTE the power to EACH FACTOR
(m5n4)3 = m15n12
RAISING A QUOTIENT TO A POWER:
DISTRIBUTE THE POWER to the numerator and the denominator
(m2/n6)3 = (m2)3/(n6)3 = m6/n18
Scientific Notation 5-4
You've had this since 6th grade!
You restate very big or very small numbers using powers of 10 in exponential form
Move the decimal so the number fits in this range: less than 10 and greater than or equal to 1 That is, 1 ≤ n < 10
Count the number of places you moved the decimal and make that your exponent
Very big numbers - exponent is positive
Very small numbers (decimals) - exponent is negative (just like a fraction!)
Remember that STANDARD notation is what you expect (the normal number)
When you multiply or divide scientific notations, use the power rules!
Just be careful that is your answer does not fit the scientific notation range, that you restate it.
TRY THIS LINK THAT TAKES YOU FROM HUGE POWERS TO
LITTLE TINY POWERS (NEGATIVE POWERS OR DECIMALS)
Try this link to practice scientific notation!
Try this link to practice multiplication of scientific notation!
Try this link to practice division!
Try this link to practice harder problems!
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