2.1 Understanding Powers and Exponents

[Pages:21]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

"1" IS ALWAYS A FACTOR!!!

ex. 1

1 ? 3 ? 3 ? 3 ? 3 = 34

exponent - "3'" is used as a factor four times. base

ex 2. 52 } power

= 1 ? 5 ? 5 "5" is used as a factor two times.

ex. 3 a3 = 1?a?a?a

"a" is used as a factor three times.

ex. 4 70

= 1 "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

2.1 continued Evaluating Powers

Find the value of the power by multiplying.

September 19, 2012

Evaluate 25

power

=2?2?2?2?2 factored form

=32

product (value)

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

Unit 1 Ch. 2 Powers and Exponents

September 19, 2012

2. 2 Order of Operations Do them in the order of top to

bottom level 1 step at a time!

P

E

M

D

A

S

Parenthesis and other grouping symbols Exponents Multiply and Divide left to right Add and Subtract left to right

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 3

= 5

grouping symbol (vinculum) division

Unit 1 Ch. 2 Powers and Exponents

September 19, 2012

2.2 Order of Operations (cont): more grouping symbols

ex. 3

(5 + 2)2 - 10

= 72 - 10

= 49 - 10

= 39

original expression parenthesis exponent subtraction

ex 4. - | 4 | + 62 ? 9 - | 4 ? 2 | -4 + 62 ? 9 - 8

original expression (grouping symbol : ab. val.)

-4 + 36 ? 9 - 8 -4 + 4 - 8

exponent division / mult

0 - 8

addition

-8

subtraction

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

Since 2 and 3 are prime numbers, these are simply brought down.

48

any two factors of 48

6 ? 8 2?3 ? 2 ? 4

6 and 8 are not prime so they are furthered factored.

4 is further factored. 2?3 ? 2 ?2 ?2

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. 1 Algebraic

original ratio

4ab2 6a2

=

prime factor and cancel

2 ? 2 ? a ? b ? b 2 ? 3 ? a ? a

products of the remaining factors

= 2b2 3a

Ex 2 Numeric

12 16

=

2 ? 2 ? 3 2 ? 2 ? 2 ? 2

=

3 4

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