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 = m6/n18
Thursday, November 4, 2010
Math 6 Honors (Period 6 and 7)
Multiplying or Dividing by a Power of Ten 3-7
We have learned that in a decimal or a whole number each place value is ten times the place value to its right.
10 ∙ 1 = 10
10 ∙ 10 = 100
10 ∙ 100 = 1000
10 ∙ 0.1 = 1
10 ∙ 0.01 = 0.1
10 ∙ 0.001 = 0.01
Notice that multiplying by ten has resulted in the decimal point being moved one place to the right and in zeros being inserted or dropped.
Multiplying by ten moves the decimal point one place to the right
10 ∙ 762 = 7620
10 ∙ 4.931 = 49.31
At the beginning of this chapter you learned about powers of ten
104 = 10 ∙10 ∙ 10 ∙10 = 10,000
We can see that multiplying by a power of 10 is the same as multiplying by 10 repeatedly.
2.64874 ∙104 = 26,387.4
Notice that we have moved the decimal point four places to the right.
Rule
To multiply a number by the nth power of ten, move the decimal point n places to the right.
Powers of ten provide a convenient way to write very large numbers. Numbers that are expressed as products of two factors
(1) a number greater than or equal to 1, but less than 10,
AND
(2) a power of ten
are said to be written in scientific notation.
We can write 'a number greater than or equal to 1, but less than 10' as an mathematical inequality 1 ≤ n < 10 To write a number in scientific notation we move the decimal point to the left until the resulting number is between 1 and 10. We then multiply this number by the power of 10, whose exponent is equal to the number of places we moved the decimal point. 4,592,000,000 in scientific notation First move the decimal point to the left to get a number between 1 and 10 4,592,000,000 the first factor in scientific notation becomes 4.592 Since the decimal point was moved 9 places, we multiply 4.592 by 109 to express the number in scientific notation
4.592 x 109 (Yes, you get to use the × symbol for multiplication .. but only for this!!
When we move a decimal point to the left, we are actually dividing by a power of ten.
Notice that in dividing by a power of 10 we move the decimal point to the left the same number of places as the exponent. Sometimes we may have to add zeros
3.1 ÷ 104 = 0.00031
Rule
To divide a number by the nth power of ten, move the decimal point n places to the left, adding zeros as necessary.
We have learned that in a decimal or a whole number each place value is ten times the place value to its right.
10 ∙ 1 = 10
10 ∙ 10 = 100
10 ∙ 100 = 1000
10 ∙ 0.1 = 1
10 ∙ 0.01 = 0.1
10 ∙ 0.001 = 0.01
Notice that multiplying by ten has resulted in the decimal point being moved one place to the right and in zeros being inserted or dropped.
Multiplying by ten moves the decimal point one place to the right
10 ∙ 762 = 7620
10 ∙ 4.931 = 49.31
At the beginning of this chapter you learned about powers of ten
104 = 10 ∙10 ∙ 10 ∙10 = 10,000
We can see that multiplying by a power of 10 is the same as multiplying by 10 repeatedly.
2.64874 ∙104 = 26,387.4
Notice that we have moved the decimal point four places to the right.
Rule
To multiply a number by the nth power of ten, move the decimal point n places to the right.
Powers of ten provide a convenient way to write very large numbers. Numbers that are expressed as products of two factors
(1) a number greater than or equal to 1, but less than 10,
AND
(2) a power of ten
are said to be written in scientific notation.
We can write 'a number greater than or equal to 1, but less than 10' as an mathematical inequality 1 ≤ n < 10 To write a number in scientific notation we move the decimal point to the left until the resulting number is between 1 and 10. We then multiply this number by the power of 10, whose exponent is equal to the number of places we moved the decimal point. 4,592,000,000 in scientific notation First move the decimal point to the left to get a number between 1 and 10 4,592,000,000 the first factor in scientific notation becomes 4.592 Since the decimal point was moved 9 places, we multiply 4.592 by 109 to express the number in scientific notation
4.592 x 109 (Yes, you get to use the × symbol for multiplication .. but only for this!!
When we move a decimal point to the left, we are actually dividing by a power of ten.
Notice that in dividing by a power of 10 we move the decimal point to the left the same number of places as the exponent. Sometimes we may have to add zeros
3.1 ÷ 104 = 0.00031
Rule
To divide a number by the nth power of ten, move the decimal point n places to the left, adding zeros as necessary.
Tuesday, November 2, 2010
Math 6 Honors (Period 6 and 7)
Adding & Subtracting Decimals 3-6
RULES TO FOLLOW:
1. Write the given numbers one above the other (STACK'EM) with the decimal points lined up!!
2. Add zeros to get the same number of decimal places and then add/sub just like whole numbers.
3. Place the decimal point in your sum/ difference in the position right under the decimal points in the given numbers.
6.47 + 340.8 + 73.523
STACK THEM!!
The use of rounded numbers to get an approximate answer is called estimation. We use estimates to check actual answers. You can see if your answer is reasonable by using estimation.
RULES TO FOLLOW:
1. Write the given numbers one above the other (STACK'EM) with the decimal points lined up!!
2. Add zeros to get the same number of decimal places and then add/sub just like whole numbers.
3. Place the decimal point in your sum/ difference in the position right under the decimal points in the given numbers.
6.47 + 340.8 + 73.523
STACK THEM!!
The use of rounded numbers to get an approximate answer is called estimation. We use estimates to check actual answers. You can see if your answer is reasonable by using estimation.
Monday, November 1, 2010
Algebra (Period 1)
Exponents: 5-1
POWER RULES:
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 = mmm
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
= m0 (by power rules for division)
By the transitive property of equality : 1 = m0
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
JUST REMEMBER
NEGATIVE POWERS MEANS THE NUMBERS ARE FRACTIONS
They're in the wrong place in the fraction
m3/m5 = m-2
m3/m5 = mmm/ mmmmm
= 1/mm
Again, by transitive property of equality:
m3/m5 = m-2 = 1/m2
Remember the rule of powers with ( )
When there is a product inside the ( ), then everything inside is to the power!
If there are no ( ), then only the variable/number right next to the power is raised to that power.
3x-2 does not equal (3x)-2
The first is 3/x2 and the second is 1/9x2
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
POWER RULES:
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 = mmm
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
= m0 (by power rules for division)
By the transitive property of equality : 1 = m0
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
JUST REMEMBER
NEGATIVE POWERS MEANS THE NUMBERS ARE FRACTIONS
They're in the wrong place in the fraction
m3/m5 = m-2
m3/m5 = mmm/ mmmmm
= 1/mm
Again, by transitive property of equality:
m3/m5 = m-2 = 1/m2
Remember the rule of powers with ( )
When there is a product inside the ( ), then everything inside is to the power!
If there are no ( ), then only the variable/number right next to the power is raised to that power.
3x-2 does not equal (3x)-2
The first is 3/x2 and the second is 1/9x2
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
Math 6 Honors (Period 6 and 7)
Rounding 3-5 (continued)
Round the following number to the designated place value:
509.690285
tenths: 509.690285
You underline the place value you are rounding to and look directly to the right. If it is 0-4 you round down; if it is 5-9 you round up 1.
so here we round to
509.7
hundredths
509.690285
becomes 509.69
hundred-thousandths
509.690285
becomes
509.69029
tens
509.690285
becomes
510
(a) What is the least whole number that satisfies the following condition?
(b) What is the greatest whole number that satisfies the following condition?
A whole number rounded to the nearest ten is 520.
Well, 515, 516, 517, 58, 519, 520, 521, 522, 523, 524 all would round to 520
so
(a) 515
(b) 524
A whole number rounded to the nearest ten is 650
(a) 645
(b) 654
A whole number rounded to the nearest hundred is 1200
(a) 1150
(b) 1249
How about these...
(a) What is the least possible amount of money that satisfies the following condition?
(b) What is the greatest possible amount?
A sum of money, rounded to the nearest dollar is $57
(a) $56.50
9b) $57.49
A sum of money rounded to the nearest ten dollars $4980
(a) $4975
(b) $4984.99
Round the following number to the designated place value:
509.690285
tenths: 509.690285
You underline the place value you are rounding to and look directly to the right. If it is 0-4 you round down; if it is 5-9 you round up 1.
so here we round to
509.7
hundredths
509.690285
becomes 509.69
hundred-thousandths
509.690285
becomes
509.69029
tens
509.690285
becomes
510
(a) What is the least whole number that satisfies the following condition?
(b) What is the greatest whole number that satisfies the following condition?
A whole number rounded to the nearest ten is 520.
Well, 515, 516, 517, 58, 519, 520, 521, 522, 523, 524 all would round to 520
so
(a) 515
(b) 524
A whole number rounded to the nearest ten is 650
(a) 645
(b) 654
A whole number rounded to the nearest hundred is 1200
(a) 1150
(b) 1249
How about these...
(a) What is the least possible amount of money that satisfies the following condition?
(b) What is the greatest possible amount?
A sum of money, rounded to the nearest dollar is $57
(a) $56.50
9b) $57.49
A sum of money rounded to the nearest ten dollars $4980
(a) $4975
(b) $4984.99
Pre Algebra (Period 2 & 4)
Decimal Review
First up: Place value
The root word in decimal is "decem" which means 10.
Each time you move in the decimal system you are multiplying or dividing the place value by 10.
If you move to the LEFT, place value is MULTIPLIED by 10. If you move to the RIGHT, place value is DIVIDED by 10 (or multiplied by 1/10...remember that dividing by 10 is the same as multiplying by its reciprocal...says our BFF, the Multiplicative Inverse Property)
The middle of the decimal system is the ones place (not the decimal!)
Place value names mirror each other to the left and to the right of the ones.
EXAMPLE: Take 1 and multiply by 10 and you get the TENS.
Take 1 and divide by 10 (or multiply by 1/10) and you get the TENTHS.
Another pattern in PLACE VALUE NAMES:
They are related to how many groups of "000" after 1,000
1 million = 1,000,000
million is from "mille" which meant 1000 and 1 million is 1000 times 1000 or 10001 bigger than 1000
1 billion = 1,000,000,000 as in 1000 times 1000 times 1000 or 10002 bigger than 1000
One more example:
1 trillion = 1,000,000,000,000 as in 10003bigger than 1000
SO TO MAKE THIS EASIER TO UNDERSTAND:
EACH NEW PLACE VALUE NAME IS COUNTING THE GROUPS OF ZEROS (000) AFTER 1000!
1 QUADrillion...QUAD means 4 so 4 groups of zeros after 1000:
1,000,000,000,000,000
AND OTHER THE OTHER SIDE OF THE DECIMAL, THE NAMES ARE THE SAME EXCEPT THEY HAVE A "TH"
What you need to do on the other side is to remember that the ones is still the starting point
on the left side of the decimal point though. So if you're looking for 1 millionth, you count 6
places in total = .000001
You see that there are only 5 zeros, not 6
1 millionth = 1 times 1/1,000,000 = .000001
Instead of writing all these zeros, we can use exponents.
Exponents for powers of 10 count the number of zeros.
So 1,000,000 = 106
and one millionth = 1/1,000,000 = 10-6
ROUNDING:
Round 0 - 4 down and 5 - 9 up
EXAMPLE: 0.34 rounded to the tenths is 0.3
0.35 rounded to the tenths is 0.4
Notice you don't add zeros at the end
If there is a 9 in the next place, then think of rounding up as adding 1, that will make the number go up to "10"
EXAMPLE: 0.96 rounded to the tenths is 1.0
Think of it as adding 1 in the tenths column which would carry to the ones.
Now there is a zero in the tenths to show the reader of the number that you rounded to the tenths and not the ones.
EXAMPLE: 24,300 rounded to the thousands is 24,000
Now you need zeros to the decimal point or you'll lose the place value!
EXAMPLE: 29,600 rounded to the thousands is 30,000
Again, if it's a 9, it will need to go up to the next column...
Think of it as adding 1
First up: Place value
The root word in decimal is "decem" which means 10.
Each time you move in the decimal system you are multiplying or dividing the place value by 10.
If you move to the LEFT, place value is MULTIPLIED by 10. If you move to the RIGHT, place value is DIVIDED by 10 (or multiplied by 1/10...remember that dividing by 10 is the same as multiplying by its reciprocal...says our BFF, the Multiplicative Inverse Property)
The middle of the decimal system is the ones place (not the decimal!)
Place value names mirror each other to the left and to the right of the ones.
EXAMPLE: Take 1 and multiply by 10 and you get the TENS.
Take 1 and divide by 10 (or multiply by 1/10) and you get the TENTHS.
Another pattern in PLACE VALUE NAMES:
They are related to how many groups of "000" after 1,000
1 million = 1,000,000
million is from "mille" which meant 1000 and 1 million is 1000 times 1000 or 10001 bigger than 1000
1 billion = 1,000,000,000 as in 1000 times 1000 times 1000 or 10002 bigger than 1000
One more example:
1 trillion = 1,000,000,000,000 as in 10003bigger than 1000
SO TO MAKE THIS EASIER TO UNDERSTAND:
EACH NEW PLACE VALUE NAME IS COUNTING THE GROUPS OF ZEROS (000) AFTER 1000!
1 QUADrillion...QUAD means 4 so 4 groups of zeros after 1000:
1,000,000,000,000,000
AND OTHER THE OTHER SIDE OF THE DECIMAL, THE NAMES ARE THE SAME EXCEPT THEY HAVE A "TH"
What you need to do on the other side is to remember that the ones is still the starting point
on the left side of the decimal point though. So if you're looking for 1 millionth, you count 6
places in total = .000001
You see that there are only 5 zeros, not 6
1 millionth = 1 times 1/1,000,000 = .000001
Instead of writing all these zeros, we can use exponents.
Exponents for powers of 10 count the number of zeros.
So 1,000,000 = 106
and one millionth = 1/1,000,000 = 10-6
ROUNDING:
Round 0 - 4 down and 5 - 9 up
EXAMPLE: 0.34 rounded to the tenths is 0.3
0.35 rounded to the tenths is 0.4
Notice you don't add zeros at the end
If there is a 9 in the next place, then think of rounding up as adding 1, that will make the number go up to "10"
EXAMPLE: 0.96 rounded to the tenths is 1.0
Think of it as adding 1 in the tenths column which would carry to the ones.
Now there is a zero in the tenths to show the reader of the number that you rounded to the tenths and not the ones.
EXAMPLE: 24,300 rounded to the thousands is 24,000
Now you need zeros to the decimal point or you'll lose the place value!
EXAMPLE: 29,600 rounded to the thousands is 30,000
Again, if it's a 9, it will need to go up to the next column...
Think of it as adding 1
Sunday, October 31, 2010
Math 6H (Period 6 & 7) Yosemite Week
Comparing Decimals 3-4
Printed Notes will be handed out in class on Monday, November 1st.
We have used number lines to compare whole numbers. Number lines can be used to show comparisons of decimals. As with whole numbers, a larger number is graphed to the right of a smaller number.
In order to compare decimals, we compare the digits in the place farthest to the left where the decimals have different digits.
Compare the following:
1. 0.64 and 0.68 since 4 < 8 then 0.64 < 0.68. 2. 2.58 and 2.62 since 5 < 6 then 2.58 < 2.62 . 3. 0.83 and 0.833 To make it easier to compare, first express 0.83 to the same number of decimal places as 0.833 0.83 = 0.830 Then compare 0.830 and 0.833 since 0 <3 Then 0.830 < 0.833. Write in order from least to greatest 4.164, 4.16, 4.163, 4.1 First, express each number to the same number of decimal places Then compare. 4.164, 4.160, 4.163, 4.100 The order of the numbers from least to greatest is 4.1, 4.16, 4.163, 4.164
Rounding 3-5
A method for rounding may be stated as follows:
Find the place to which you wish to round, mark it with an underline ___
Look at the digit to the right.
If the digit to the right is 5 or greater, add 1 to the marked digit.
If the digit to the right is less than 5, leave the marked digit unchanged.
Replace each digit to the right of the marked place with a 0
Round 32,567 to (a) the nearest ten thousand,
(b) the nearest thousand,
(c) the nearest hundred, and
(d) the nearest ten
(a)32, 567: since 2 is less than 5, we leave the 3 unchanged, and replace 2, 5, 6, and 7 with zeros
30,000
(b) 32,567: since the digit to the right of 2 is 5, we add a 1 to 2 and get 3 and we replace 5, 6, and 7 with zeros
33,000
(c)32,567: since 6 is greater than 5, we add 1 to 5 and replace 6 and 7 with zeros
32,600
(d) 32,567: since 7 is greater than 5, we add 1 to 6 and replace 7 with a zero
32,570
A similar method of rounding can be used with decimals.
The difference between the two methods is that when rounding decimals, we do not have to replace the dropped digits with zeros.
Round 4.8637 to
(a) the nearest thousandth,
(b) the nearest hundredth,
(c) the nearest tenth, and
(d) the nearest unit
a. 4.8637: Since 7 is greater than 5, we add 1 to 3 --get 4 & drop the 7
4.864
b. 4.8637: Since 3 is less than 5, we leave 6 unchanged and drop 3 & 7
4.86
c. 4.8637: Since 6 is greater than 5, we add 1 to 8 and drop 6,3, &7
4.9
d. 4.8637: Since 8 is greater than 5, we add 1 to 4 and drop 8, 6, 3, & 7
5
Printed Notes will be handed out in class on Monday, November 1st.
We have used number lines to compare whole numbers. Number lines can be used to show comparisons of decimals. As with whole numbers, a larger number is graphed to the right of a smaller number.
In order to compare decimals, we compare the digits in the place farthest to the left where the decimals have different digits.
Compare the following:
1. 0.64 and 0.68 since 4 < 8 then 0.64 < 0.68. 2. 2.58 and 2.62 since 5 < 6 then 2.58 < 2.62 . 3. 0.83 and 0.833 To make it easier to compare, first express 0.83 to the same number of decimal places as 0.833 0.83 = 0.830 Then compare 0.830 and 0.833 since 0 <3 Then 0.830 < 0.833. Write in order from least to greatest 4.164, 4.16, 4.163, 4.1 First, express each number to the same number of decimal places Then compare. 4.164, 4.160, 4.163, 4.100 The order of the numbers from least to greatest is 4.1, 4.16, 4.163, 4.164
Rounding 3-5
A method for rounding may be stated as follows:
Find the place to which you wish to round, mark it with an underline ___
Look at the digit to the right.
If the digit to the right is 5 or greater, add 1 to the marked digit.
If the digit to the right is less than 5, leave the marked digit unchanged.
Replace each digit to the right of the marked place with a 0
Round 32,567 to (a) the nearest ten thousand,
(b) the nearest thousand,
(c) the nearest hundred, and
(d) the nearest ten
(a)32, 567: since 2 is less than 5, we leave the 3 unchanged, and replace 2, 5, 6, and 7 with zeros
30,000
(b) 32,567: since the digit to the right of 2 is 5, we add a 1 to 2 and get 3 and we replace 5, 6, and 7 with zeros
33,000
(c)32,567: since 6 is greater than 5, we add 1 to 5 and replace 6 and 7 with zeros
32,600
(d) 32,567: since 7 is greater than 5, we add 1 to 6 and replace 7 with a zero
32,570
A similar method of rounding can be used with decimals.
The difference between the two methods is that when rounding decimals, we do not have to replace the dropped digits with zeros.
Round 4.8637 to
(a) the nearest thousandth,
(b) the nearest hundredth,
(c) the nearest tenth, and
(d) the nearest unit
a. 4.8637: Since 7 is greater than 5, we add 1 to 3 --get 4 & drop the 7
4.864
b. 4.8637: Since 3 is less than 5, we leave 6 unchanged and drop 3 & 7
4.86
c. 4.8637: Since 6 is greater than 5, we add 1 to 8 and drop 6,3, &7
4.9
d. 4.8637: Since 8 is greater than 5, we add 1 to 4 and drop 8, 6, 3, & 7
5
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