2.1 understanding powers and exponents 1 ch. 2...unit 1 ch. 2 powers and exponents september 19,...

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"


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)


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!


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)


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


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


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


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


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


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.


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


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.


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


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


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


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


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


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

= ?


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....


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!)


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

2.1 understanding powers and exponents 1 ch. 2...unit 1 ch. 2 powers and exponents september 19,...

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