Exponents: 5-1
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!!!!
-2^5 = -32
-2^4 = -16
MULTIPLYING Powers with LIKE BASES:
Simply ADD THE POWERS
m^5m^3 = m^8
You can check this by EXPANDING:
(mmmmm)(mmm) = m^8
DIVIDING Powers with LIKE BASES:
Simply SUBTRACT the POWERS
m^8/ m^5 = m^3
Again, you can check this by EXPANDING:
mmmmmmmm/mmmmm
ZERO POWERS:
Anything to the zero power = 1
(except zero to the zero power is undefined)
Proof of this was given in class:
By the transitive property of equality : 1 = m^0
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:
m^3/m^5 = m^-2
m^3/m^5 = m^-2
mmm/mmmmm 1/mm
Again, by transitive property of equality:
m^3/m^5 = m^-2 = 1/m^2
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 on Exponents: 5-2
POWER to another POWER
Multiply the POWERS
(m^5)^3 = m^15
to check EXPAND it out (m^5) (m^5) (m^5) = m^15
PRODUCT to a POWER
DISTRIBUTE the power to EACH FACTOR
(m^5n^4)^3 = m^15n^12
RAISING a QUOTIENT to a POWER
distribute the power to the numerator AND to the denominator
Monday, December 1, 2008
Math 6 H Periods 1, 6 & 7
Review of Sections 5.1- 5.3
You know that 60 can be written as the product of 5 and 12. 5 and 12 are whole number FACTORS of 60.
A number is said to be divisible by its whole number factors if the remainder is 0.
A multiple of a whole number is the product of that number and any other whole number ( including zero-- because zero is a whole number)
You can find the multiples of any given whole number by multiplying that number by 0, 1, 2, 3, 4...
Any multiple of 2 is called an even number. A whole number that is not an even number is called an odd number.
Zero is an even number!!
Tests for Divisibility
A Number is Divisible by:
2 if is even
3 if the SUM of its digits is a multiple of 3
4 if the last two digits in the number represent a multiple of 4
5 if its last digit is either a 0 or a 5
6 if the number is divisible by 2 AND 3
8 if the last three digits in the number represent a multiple of 8
9 if the SUM of its digits is a multiple of 9
0 if the last digit is a 0.
A perfect number is one that is the sum of all its factors except itself. The smallest perfect number is 6 , since 6 = 1 + 2 + 4. What would be the next perfect number?
Try this and let me know for extra credit... email me your response.
Square Numbers and Square Roots 5-3
Numbers such as 1, 4, 9, 16, 25 are called perfect squares or square numbers.
1 = 1 times 1
4 = 2 times 2
9 = 3 times 3 and so on
One of the two equal factors of a square number is called the square root of that number. To denote the square root of a number we use a radical sign. √ it looks something like a check mark with a line extending. See our textbook for better examples. ( Page 157)
We can use our knowledge of perfect squares to find the lengths of sides of squares. For example IF the Area of a square is 49 cm squared. We can find the length of each side using the formula for the area of a square. A = lw but in this case it is also A = s^2, which is read "Area equals side squared."
so we know 49 = s^2. so the square root (SQRT) of 49... hmmm.. we know 7 times 7 = 49 so s = 7 cm. Notice the label isn't squared- we are talking about linear measurements with each side-- only the AREA is squared.
Positive SQ RT are also called principal square roots.
negative square roots exist also. For example the SQRT of 81 is 9 but it could also be -9 because (-9) times (-9) also = 81
We are focusing on positive SQ RTs in this chapter.
Memorize the Perfect Squares and corresponding SQ RTs up to 30.
KNOW the rule for squaring any number that ends in 5.
See me if you forgotten the rule!!
Check out these websites for how to calculate a SQ RT without a calculator
http://mathforum.org/library/drmath/sets/mid_square_roots.html
http://www.homeschoolmath.net/teaching/square-root-algorithm.php
http://www.nist.gov/dads/HTML/squareRoot.html
You know that 60 can be written as the product of 5 and 12. 5 and 12 are whole number FACTORS of 60.
A number is said to be divisible by its whole number factors if the remainder is 0.
A multiple of a whole number is the product of that number and any other whole number ( including zero-- because zero is a whole number)
You can find the multiples of any given whole number by multiplying that number by 0, 1, 2, 3, 4...
Any multiple of 2 is called an even number. A whole number that is not an even number is called an odd number.
Zero is an even number!!
Tests for Divisibility
A Number is Divisible by:
2 if is even
3 if the SUM of its digits is a multiple of 3
4 if the last two digits in the number represent a multiple of 4
5 if its last digit is either a 0 or a 5
6 if the number is divisible by 2 AND 3
8 if the last three digits in the number represent a multiple of 8
9 if the SUM of its digits is a multiple of 9
0 if the last digit is a 0.
A perfect number is one that is the sum of all its factors except itself. The smallest perfect number is 6 , since 6 = 1 + 2 + 4. What would be the next perfect number?
Try this and let me know for extra credit... email me your response.
Square Numbers and Square Roots 5-3
Numbers such as 1, 4, 9, 16, 25 are called perfect squares or square numbers.
1 = 1 times 1
4 = 2 times 2
9 = 3 times 3 and so on
One of the two equal factors of a square number is called the square root of that number. To denote the square root of a number we use a radical sign. √ it looks something like a check mark with a line extending. See our textbook for better examples. ( Page 157)
We can use our knowledge of perfect squares to find the lengths of sides of squares. For example IF the Area of a square is 49 cm squared. We can find the length of each side using the formula for the area of a square. A = lw but in this case it is also A = s^2, which is read "Area equals side squared."
so we know 49 = s^2. so the square root (SQRT) of 49... hmmm.. we know 7 times 7 = 49 so s = 7 cm. Notice the label isn't squared- we are talking about linear measurements with each side-- only the AREA is squared.
Positive SQ RT are also called principal square roots.
negative square roots exist also. For example the SQRT of 81 is 9 but it could also be -9 because (-9) times (-9) also = 81
We are focusing on positive SQ RTs in this chapter.
Memorize the Perfect Squares and corresponding SQ RTs up to 30.
KNOW the rule for squaring any number that ends in 5.
See me if you forgotten the rule!!
Check out these websites for how to calculate a SQ RT without a calculator
http://mathforum.org/library/drmath/sets/mid_square_roots.html
http://www.homeschoolmath.net/teaching/square-root-algorithm.php
http://www.nist.gov/dads/HTML/squareRoot.html
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