6.1 Multiplication with Exponents and Scientific Notation

6.2 Division with Exponents and Negative Exponents

6.3 Operations with Monomials

6.1 Multiplication with Exponents

Recall that an exponent is a number placed above and to the right of another number. The other number is referred to as the BASE.

xbBASE

EXPONENT

• Exponents are sometimes referred to as powers. “Special powers like ‘2’ have phrases like ‘squared’ and ‘3’ has the phrase ‘cubed’” Everything else we would say “the base raised to that exponent”

KNOW THAT IF A POWER IS NOT WRITTEN THEN IT IS AN UNDERSTOOD ONE “1”. AS YOU BEGIN DOING

THIS WORK IT MAY BE BENEFICIAL AT TIMES TO PLACE A 1 WHEN IT IS NOT WRITTEN.

1

1

5 5

x x

Evaluating ExponentsExponents simply tell us how many times we are multiplying the base by itself.

So some examples: 53 means that we are multiplying 5 by itself 3 times. 5 x 5 x 5.

EVALUATING tells us to find the equivalent value of 53 which in this case would be 125.

Important Notation

• An exponent is intended to be applied only to the base that it is directly behind. Therefore if we have something like the following…

• The power of 4 only gets applied to the three.So an equivalent statement would be 12 x 3 x 3 x 3 x 3 = 972

412 3

• Now if we were to incorporate a set of parentheses in the same expression it changes the way we think about what the power gets applied to.

• Now the idea is that the power of 4 is applied to the entire quantity inside the ( ) and not just the 3. So this would be (36)4=1,679,616

4(12 3)

• The biggest mistake that people make is in regards to negative numbers and exponents.

• Recall that an even amount of negatives will always give a positive product and an odd number of negatives will yield a negative result. Take the two examples below

4( 2)In this example we are viewing the -2 as an entirequantity so that entire term is raised to the 4th power. we would get -2 x -2 x -2 x -2

42In this example because there Are no ( ) the power of 4 is only To be applied to the thing that itIs directly behind which is the 2. NOT THE --

Logic behind

• An alternative way of thinking about this expression is to rewrite it as

42

42 41 2 Thus we now can visualize that the power of 4is only behind the 2 and not the -1. Many people will not rewrite the problem with this notation but understand that this is why the – does not get the powerof 4 applied to it.

Simplify / Evaluate3

3

3

2

4

3

.) ( 3)

.

.)

) (

.) 3

4.)

3

2

2)

.) 5a

e

c

d

b

f

Product Property• Occasionally we will want to multiply bases that are the same but

both have exponents.

• The idea is that we can expand this to

5 x 5 x 5 x 5 x 5

How many 5’s do we have? Can you look at the original problem and determine a

rule that would apply?

235 5

• So simply stated when we are multiplying bases together that are “like bases” then we add the exponents.

r s r sb b b

Simplify (leave answers in terms of base with one exponent)

4 3

4 6 1

2 4 3

.) 3 3

.) 2 2 2

.) 3 3 3 3 3

.)

.) t s w

a

b

c

d x x x

e r r r

Power Property (sometimes called power to a power)

• Lets use what we already know about exponents to develop a rule for this.

Focus on the red exponent first, what does that mean?

Now if multiplying like bases what do you do with exponents?

2 3(5 )2 2 2(5 )(5 )(5 )

65

So what is the rule?

• When a power is raised to another power we simply keep the same base and multiply the powers.

( )r s r sb b

Simplify

4 2

3 4

.) (3 )

.) ( )

.) ( )x y

a

b x

c w

Distributive Property of Exponents• This property comes in handy when an exponent is

being applied to a quantity that contains two or more numbers or variables.

• The whole idea is based on what we already know.

• But that also could be written as

• Which can be written as

• Which we can then change back to

3(4 )y

4 4 4y y y

4 4 4y y y

4 4 4 y y y 3 34 y

• So basically what we have done is distributed the power of 3 to all parts inside the parentheses. Remember distribute is a word that implies to “give to all” but in mathematics we “give to all through multiplication”

• So a rule for this would look like the following

( )s t r sr tra b a bIn the previous example s and t were both 1.

Simplify

23 2

3 2 7

3 2 3 5 3

3 2 4

1.)

3

.) ( ) (2 )

.) (3 ) (2 )

.) (2( ) )

a x y m

b w w

c x y xy m

d x

Unless you have nested parentheses then you want to apply the power to power or distributive properties first. Understand that the distributive property is simply an extension of the power to a power rule.

Scientific Notation

• Scientific notation is a technique for writing very large or very minute values. When numbers incorporate many place values to either side of the decimal, calculations can get messy. Scientific notation is an attempt to alleviate some of that “messiness”

• Scientific notation will always look like the following.

• Where n is always a number between 1 and 10. And r is an integer (…-3, -2, -1, 0, 1, 2, 3…)

10rn

• When dealing with scientific notation and really large or small values we will want to first locate the current decimal point. Then the goal is to move that decimal point so that it is between the first two numbers that are non-zero values.

EXAMPLES

454,000,000

213,000

5,321,000,000,000

6.2 Division with Exponents and Negative Exponents

• When multiplying like bases we ADDED exponents.

• When dividing like bases we will now SUBTRACT exponents.

• The idea is that you can expand the top and bottom to the following

• And then reduce. Leaving just 4. That can be more easily done by subtracting 3-2.

3

2

4

4

4 4 4

4 4

• When subtracting (at first at least) we will always subtract so that our exponents are positive (bigger-smaller) and then wherever the larger exponent began is where our new total will go.

2

5

5

5

8

5

4

4

Negative Exponents• If exponents are negative it means that we have a

fraction as an answer. For instance if we would have taken the following and subtracted “top-bottom”

2

5

5

5We would have gotten …

But we do not like negative exponents in answers, so we transition The negative exponent and its base to the denominator and it becomes

A positive exponent.

35

Practice re-writing negative exponents so they are positive.

2

3

5

4

.)3

.)4

.)4

.)3

a

b

c x

d xy

• Now from time-to-time we may end up with a negative exponent that shows up in the denominator of the fraction, if this occurs we take the base and exponent that is negative and move it to the numerator and make it positive. Examples.

2

2

3

1.)42

.)

1.)3

a

bx

cab

• We do not like negative exponents in our answers, whenever you have negative exponents to make them positive exponents you must take the exponent and the base that it belongs to and move them on the other side of the division bar. (if you follow this rule it doesn’t matter which order you subtract exponents)10

4

5

7

21

25

.)

.)

2.)2

xax

xbx

c

Expanded Distributive Property• If there is division occurring but the whole quotient is

raised to a power, you must distribute that power to every exponent in the numerator, as well as every exponent in the denominator.

2

3

7

3

.)5

2.)

3.)2

xa

bx

c

• Any base raised to the zero power is ALWAYS simplified to equal 1.

• Here is an example of why, use your rules to evaluate. But first make a determination of what you get when you divide something by itself. 3

3

5

5

Combining Properties

• A general practice, because its needed in other areas, is to take all of these properties and force you to work with them inside one problem. We will do several examples.

• Get rid of parentheses first. So apply power to powers first if present. Do division last, if present. (ALWAYS WRITE ANSWERS WITH POSITIVE EXPONENTS)

6

3 4.)( )

xa

x

23

4

2.)

xb

x

28

3.)y

cy

24.) 2d x

6 2.)e x x

5 2 4

3 2

( ).)( )

a af

a

Scientific Notation for Small Numbers

Here the numbers will have a decimal place out front followed by several zeroes and then some non-zero values. We will want to move the decimal place so that it stops between the first two non-zero values. Count the decimal places moved and that will be your exponent. However because we moved the decimal to the right, the exponent will be negative.

Examples.00004324

.0002

.0000982

.0032

*reminder explain why this works on the chalkboardusing negative exponents with the base of 10*

6.3 Operations with Monomials

• A monomial is a one-term expression that is either constant (number “non-changing”) or the product of a constant and one or more variables raised to whole number exponents.

• Here are some examples

2

3

15x

x y

4 2 4

2

2 3

49

23

3

4

x y z

x y

a b

• The number associated with monomials is called the coefficient. A monomials are also referred to as “terms”

• Rules for operating with monomials. – Apply all rules that we learned about exponents– Can only multiply / divide like terms (same base)– Coefficients only work with coefficients

• Divide coefficients and multiply coefficients

• Commutative and Associative properties are often useful when simplifying.

Examples

Examples4 3

2

27

9

x y

xy

Examples

Examples

Examples

Multiplying and Dividing Numbers in Scientific Notation

Multiplying multiply the two numbers out in front, and then add the exponents that go with 10. Re-write abiding by scientific notation rules

Dividing Divide the two numbers out in front, and then subtract the exponents that go with 10. Re-write abiding by scientific notation rules.

Examples

Examples

Examples

Two monomials with the exact same variable part (same variables raised to the exact same powers) are called similar or like terms.

When adding and subtracting similar/like terms you add/subtract coefficients and that is all