Unit 1 Ch. 2 Powers and Exponents September 19, 2012
2.1 Understanding Powers and Exponents
power- a combination of a base and an exponent
base- the number or variable in a power being used as a factor
exponent- the superscripted number that tells you how many times the base is used as a factor
factor- a number or variable being multiplied
ex. 1 1 • 3 • 3 • 3 • 3 = 34 exponent - "3'" is used as a factor four times.base
ex 2. 52 }
= 1 • 5 • 5 "5" is used as a factor two times.
power
ex. 3 a3
= 1 • a • a • a "a" is used as a factor three times.
ex. 4 70
= 1
"1" IS ALWAYS A FACTOR!!!
"7" is used as a factor zero times!!!
Therefore:ANY NUMBER TO THE ZERO POWER IS EQUAL TO "1"
Unit 1 Ch. 2 Powers and Exponents September 19, 2012
2.1 continued Evaluating Powers
To evaluate variable powers, substitute in the given value for the variable then simplify.
ex. If a = 5
a3 original expression
= 53 substitute
= 5 • 5 • 5 factored form= 125 simplify/product
Find the value of the power by multiplying.
Evaluate 25 power
=2•2•2•2•2 factored form
=32 product (value)
Unit 1 Ch. 2 Powers and Exponents September 19, 2012
2. 2 Order of Operations
P
E
M D
A S
Parenthesis and other grouping symbols
Exponents
Multiply and Divide left to right
Add and Subtract left to right
Do them in the order of top to bottom level 1 step at a time!
Unit 1 Ch. 2 Powers and Exponents September 19, 2012
2.2 cont Evaluating a variable expression using Order of Operations
Substitute in the values for the variables then follow the Order of Operations
ex. if a = 4 evaluate
3 + 2 • a2 original expression
= 3 + 2 • 42 substitute
= 3 + 2 • (4 • 4) evaluate the exponent= 3 + 2 • 16
= 3 + 32 product of 2 • 16 ( multiply)
= 35 sum of numbers (add)
Unit 1 Ch. 2 Powers and Exponents September 19, 2012
2.2 cont. Order of Operations: Using Grouping Symbols
Parenthesis, brackets, and vinculum (division/fraction bar) are all grouping symbols.
ex1. 4 ( 6 - 3 ) original expression
= 4 ( 3 ) parenthesis
= 12 multiply
ex. 2. 10 + 5 7 - 4 original expression
= 15 grouping symbol (vinculum) 3
= 5 division
Unit 1 Ch. 2 Powers and Exponents September 19, 2012
ex. 3 (5 + 2)2 - 10 original expression
= 72 - 10 parenthesis
= 49 - 10 exponent
= 39 subtraction
ex 4. - | 4 | + 62 ÷ 9 - | 4 • 2 | original expression
(grouping symbol : ab. val.)
exponent
division / mult
addition
subtraction
-4 + 62 ÷ 9 - 8
-4 + 36 ÷ 9 - 8
-4 + 4 - 8
0 - 8
-8
2.2 Order of Operations (cont): more grouping symbols
Unit 1 Ch. 2 Powers and Exponents September 19, 2012
2.3 Monomials and Powers: 2.3a Prime and Algebraic Factorization
Factor Trees help to organize the factors as you work your way to the primes.
Prime factor 48
48
6 • 8
2 • 3 • 2 • 4
2 • 3 • 2 • 2 • 2
any two factors of 48
6 and 8 are not prime so they are furthered factored.
4 is further factored.
Since 2 and 3 are prime numbers, these are simply brought down.
The product of the prime factors should be equal to the starting number.
The prime factorization can be expressed using exponents 2•3•2•2•2 = 24 • 3
Algebraic factorization is done the same way!
20a2b3
20 • a2 • b3
4 • 5 • a • a • b • b • b
2 • 2 • 5 • a • a • b • b • b
Unit 1 Ch. 2 Powers and Exponents September 19, 2012
2.3b Simplifying Numeric and Algebraic Ratios
1. Write the numerator in prime factored form.
2. Write the denominator in prime factored form.
3. Cancel all expressions of 1.
4. Find the product of factors remaining in numerator.
5. Find the product of the factors remaining in the denominator.
Ex. 1Algebraic
4ab2
6a2 = 2 • 2 • a • b • b2 • 3 • a • a = 2b2
3a
originalratio
prime factorand cancel
products of the remaining factors
12 2 • 2 • 3 316 2 • 2 • 2 • 2 4==
Ex 2Numeric
Unit 1 Ch. 2 Powers and Exponents September 19, 2012
2.3b continued Simplifying a monomial expression:
1. Expand the expression
2. cancel expressions of 1
ex2) (2x3)
(-3x)
ex) = 2•x•x•x (-3)•x
= 2•x•x•x -1•3•x
3. Write the remaining ratio.
=2x2 OR - 2 x2
-3 3
Unit 1 Ch. 2 Powers and Exponents September 19, 2012
2.3c Algebraic Understanding: Rules of Exponents
Think: a3• a2
= (a•a•a) • (a•a) each in factored form= a•a•a•a•a Since they are all the same base (factor),
you see how many "a" factors you have in all.
This is the same as adding the exponents.
= a3+2
= a5 write product as a power
Ex2) a2• b2 •a2
= a2•a2•b2 commute like-bases
= a2+2 • b2 add the exponents of the like-bases.
= a4 • b2 *write product as powers
Ask: How many factors of "a" altogether? 2+2=4
How may factors of "b" altogether? 2
So, you can add the "same base" exponents to count how many of each factor there is.
This is called the Product of Powers rule!
Since a3 and a2 have the same BASE, it means you are multiplying by more of the same factor.
Unit 1 Ch. 2 Powers and Exponents September 19, 2012
Simplifying fractions and algebraic ratios
1. Factor the numerator.2. Factor the denominator.3. Cancel expressions of 1.
ex) a5 = a•a•a•a•aa3 a•a•a
= a2 = a2
1
ex) 6x3y2 = 2•3•x•x•x•y•y = 2x2y = 2x2y3xy 3•x•y 1
The Quotient of Powers rule: If the bases are the same, subtract the exponent in the denominator from the exponent in the numerator.
2.3 c Exponent Rule: Quotient of Powers
Since, there are 5 "a"s in the numerator, they can cancel all 3 of the "a" in the denominator. This leaves 2 "a"s in the numerator. So, you are subtracting 3 "a"s from the numerator: a5-3
a5
a3= a5-3 = a2
Unit 1 Ch. 2 Powers and Exponents September 19, 2012
2.3c Exponent Rule: Power of Powers
ex. (3a2)3
This means
(3a2)3 NOTICE: (3a2) is the base!
=(3a2)(3a2)(3a2) (3a2) is used as a factor 3 times!
Commute like-base factors= 31•31•31•a2•a2•a2
=31 • a2 • 31 • a2 • 31 • a2 separate factors
= 31•3 • a2•3 apply Power of Powers31 three times and a2 three times.
= 27a6
Power of Powers Rule states when you have a power raised to a power, multiply the exponent of the entire base by each factor in the base.
Unit 1 Ch. 2 Powers and Exponents September 19, 2012
2.4 Negative and Zero Exponents
Definition of Negative Exponents: For any integer (n) and any number (a) except zero.
a-n = 1 an
ex) 5-2 = 152
You can write negative powers as positive powers using this knowledge.
ex) a-3 = 1a3
ex) 2 = 2x-4
x4You can write fraction powers as a string of powers by using the reverse of this!
Definition of negative exponents
ex) 35 = 35a2
a-2Definition of negative exponents
** Think of the negative exponent as saying, "Move the location of this power (from top to bottom or from bottom to top) and change the sign of the exponent.
changed location of just the power and change sign of exponent.
changed location of power and sign of exponent.
4 4
Unit 1 Ch. 2 Powers and Exponents September 19, 2012
2.4 cont. Simplifying numerical and variable expressions with negative exponents
1. Look for any negative exponents and relocate them so that all exponents are positive.
2. Simplify the expression using canceling of expressions of one.
3. Write the answer in the form requested (either with or without negative exponents.)
ex) -2a2b-3 = -2•a2= -1•2•a•a = -1a 4a 4ab3 2•2•a•b•b•b 2b3
Note: " -2 " does not relocate because it is not a negative exponent. Remember- the negative in this case is a factor of -1. -2=(-1•2)
Product of Powers Rule with Negative exponents:
ex. a2 b3 a2-5 • b3-2 = a-3 •b1 OR ba5 b2 a3
=Product of Powers
Unit 1 Ch. 2 Powers and Exponents September 19, 2012
2.5 Scientific Notation: Use of Powers in Scientific Notation
Scientific notation is expression a number as the product of a number between 1 and 10 and a power of 10.
1. If we want to write a number in scientific notation, first we use the digits but place the decimal to create a number between 1 and 10.
2. Next, we determine the power of ten by asking, "Does this number have to increase (multiply by a positive exponent) or decrease(multiply by a negative exponent to be equal to the original number?"
ex. Write 345,000 in scientific notation
345,000 = 3.45 X 10?
use digit to create number between 1 and 10
3.45 has to increase to get back to the original number, so multiply by a positive poser of 10.
3. Then, ask, "How many places must the digits move to get back to the original position?"
The 3 must go from the ones place to the 100,000 place, so it must increase by 5 place. Therefore, the power is 105
345,000 = 3.45 X 105
Unit 1 Ch. 2 Powers and Exponents September 19, 2012
2.45 X 104
The digits in the number will increase by 4 (the exponent) place values. So, the 2 in 2.45 is in the ones place. We look at the exponent of 104
and increase the 2 four place values. Therefore, the 2 ends up in the ten-thousands place with the other digits following.
If we want to write a number in standard form which is in scientific notation, we look at the exponent to determine if the number is going to increase (positive exponent) or decrease (negative exponent) in value, and by how many place values (exponent number)
increase four places
decrease three places3.52 X 10-3
2 4 5 0 0._ _ _ _ 2.45
So, 2.45 X 104 = 24500 in standard form
_ . _ _ 3 4 5= .003453 . 4 5
in standard form.
Ex. 2
Ex. 1
2.5 Scientific Notation cont: Scientific to Standard Form
Unit 1 Ch. 2 Powers and Exponents September 19, 2012
2.6 a Factoring and Roots
The of the word "root". It is as the base of a plant.
Root in math is the base of a power. The exponent tells us how many time the base is used as a factor.
So, if we are talking "square root" the exponent is 2. Therefore, we are looking for the base of a 2nd power.
25
means what is the (root) base (n) of the power n2= 25 ?
√
Steps:1. Prime factor the number.
2. Use the prime factors to create 2 (or whatever the exponent is worth) equal factors. That factor is the base or root!!
25
5 • 5
square root so 2 containters
5 5There is a factor of 5for each container, therefore the root is 5.
25√ = 5
Unit 1 Ch. 2 Powers and Exponents September 19, 2012
2.6a continued
If there are left over prime factors leave them under the radical.
√ 50
50
2•5•5
5 5leftover
2
Therefore the answer is
√ 50 = 5 √ 2
= ?
Unit 1 Ch. 2 Powers and Exponents September 19, 2012
2.6b Estimating Square Roots
Identify the perfect squares:
roots perfect squares
1 1•1 = 12 2•2 = 43 3•3 = 94 4•4 = 165 5•5 = 256 6•6 = 367 7•7 = 498 8•8 = 649 9•9 = 8110 10•10= 100
AND SO ON....
Unit 1 Ch. 2 Powers and Exponents September 19, 2012
2.6b continued To estimate the square root of a number that is not a perfect square:
1. Determine which perfect squares the number is between.
2. Compare the number to the midpoint between the perfect squares and determine which perfect square it is closer to.
3. Look at what roots it is between and estimate a root which is approximately the same distant from the roots as the number is from the perfect squares.
0 1 4 9 16 25
ROOTS
SQUARES
EX: Approximate the sq. root of 10?
10
3.2
1. 10 is between the perfect squares 9 and 16.
2. 10 is much closer to 9 than 16 (13 is half way!)
3. 10's root must be much closer to 3 than to 4. So my estimate is 3.2!
( Bar is interactive. Slide to location of number!)
Unit 1 Ch. 2 Powers and Exponents September 19, 2012
Same steps! 144
2•2•2•2•3•3
2.6c Other roots: Cubed root, 4th root,... Remember leftovers stay under the radical sign!
√ 144
√
√
3144
4144
2•2•2•2•3•3
2•2•2•2•3•3
2•2•2•2•3•3
2•2•3 2•2•3 no leftovers
2 22
2 2 2 2
2 • 3 • 3 leftover
3•3 leftover
12√ 144 =
√3
144 = 2 18√
√4 144 = 2 9√4
3