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Laws of Exponents. 7-2 through 7-4. What we call expanded notation. What we call expanded notation. What we call expanded notation. Putting it all together…. Putting it all together… =3  3  3  3  3  3. Putting it all together… =3  3  3  3  3  3

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Laws of Exponents

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Laws of exponents

Laws of Exponents

7-2 through 7-4


Laws of exponents

  • What we call expanded notation


Laws of exponents

  • What we call expanded notation


Laws of exponents

  • What we call expanded notation


Laws of exponents

  • Putting it all together…


Laws of exponents

  • Putting it all together…

    • =3 3 3 3 33


Laws of exponents

  • Putting it all together…

    • =3 3 3 3 33

    • All told, how many 3’s?


Laws of exponents

  • 6, so our final answer, in exponential notation, is…

    • =3 3 3 3 33

    • All told, how many 3’s?


Laws of exponents

  • 6, so our final answer, in exponential notation, is…

    • =

    • All told, how many 3’s?


Laws of exponents

  • Is there a quicker way?

    • =

    • All told, how many 3’s?


Laws of exponents

  • Is there a quicker way?

    • =

    • All told, how many 3’s?

      Absolutely; we can gain the same answer by


Laws of exponents

  • Is there a quicker way?

    • =

    • All told, how many 3’s?

      Absolutely; we can gain the same answer by adding exponents.


Giving us our 1 st law

Giving us our 1st law:

  • Is there a quicker way?

    • =

    • All told, how many 3’s?

      Absolutely; we can gain the same answer by adding exponents.


Giving us our 1 st law1

Giving us our 1st law:


Giving us our 1 st law2

Giving us our 1st law:

-When we multiply powers with the same base, we add their individual exponents.


Examples

Examples:


Examples1

Examples:


Examples2

Examples:


Examples3

Examples:


Make sure to differentiate between

Make sure to differentiate between…


Make sure to differentiate between1

Make sure to differentiate between…

Exponents and Coefficients


Make sure to differentiate between2

Make sure to differentiate between…

Exponents and Coefficients

Students will often get mixed up and apply the wrong operation to the problem.


Example of what i mean

Example of what I mean…


Example of what i mean1

Example of what I mean…

4(9) = 36


Example of what i mean2

Example of what I mean…

4(9) = 36


Example of what i mean3

Example of what I mean…

4(9) = 36


Example of what i mean4

Example of what I mean…

4(9) = 36


Example of what i mean5

Example of what I mean…

=


Example of what i mean6

Example of what I mean…

=


Example of what i mean7

Example of what I mean…

=

6(9) = 54


Example of what i mean8

Example of what I mean…

=

6(9) = 54


You try

You try…


You try1

You try…


First steps into

First steps into…


First steps into1

First steps into…

Scientific Notation!!!


First steps into2

First steps into…


First steps into3

First steps into…

1 < |a| < 10


First steps into4

First steps into…

1 < |a| < 10

n is an integer.


Examples4

Examples:

256,000

0.0041


Examples5

Examples:

256,000

0.0041


Examples6

Examples:

256,000

0.0041


Keep in mind

Keep in mind:


Keep in mind1

Keep in mind:

  • a is negative when the original number is negative.


Keep in mind2

Keep in mind:

  • a is negative when the original number is negative.

  • With small decimals, the absolute value of the exponent is equal to the # of zeroes, if you include a lead zero before the decimal place.


Real world application

Real world application:


Real world application1

Real world application:

  • At 20 Celsius, one of water has a mass of about 9.98  grams.


Real world application2

Real world application:

  • At 20 Celsius, one of water has a mass of about 9.98  grams. Each gram of water contains about 3.34  molecules of water


Real world application3

Real world application:

  • At 20 Celsius, one of water has a mass of about 9.98  grams. Each gram of water contains about 3.34  molecules of water. How many molecules of water are contained in a swimming pool containing 200 of water?


Solution dimensional analysis

Solution: Dimensional Analysis

O =


Solution dimensional analysis1

Solution: Dimensional Analysis

O =


Solution dimensional analysis2

Solution: Dimensional Analysis

O =


Solution dimensional analysis3

Solution: Dimensional Analysis

O =


Solution dimensional analysis4

Solution: Dimensional Analysis

O =


Solution dimensional analysis5

Solution: Dimensional Analysis

O =


Solution dimensional analysis6

Solution: Dimensional Analysis

O =


Next law

Next Law


Next law1

Next Law

POWER LAW!!!


Next law2

Next Law

POWER LAW!!!

Raising an exponent inside parentheses to another exponent.


Next law3

Next Law

POWER LAW!!!

Example:


Next law4

Next Law

POWER LAW!!!

Example:


Expanded

Expanded!

POWER LAW!!!

Example:


Expanded1

Expanded!

POWER LAW!!!

Example: =


Expanded2

Expanded!

Return to the product law

Example: =


Expanded3

Expanded!

Return to the product law

Example: =


Expanded4

Expanded!

Which I can obtain much faster by…

Example: =


Multiplication

Multiplication

Which I can obtain much faster by…

Example: =


Power law

Power Law:


Power law1

Power Law:


Power law2

Power Law:


Power law3

Power Law:

When raising an exponent to another exponent, we multiply the individual exponents


Examples7

Examples:


Examples8

Examples:


Examples9

Examples:


Examples10

Examples:


Examples11

Examples:


One problem

One problem:


One problem1

One problem:


One problem2

One problem:


Combining it with the product law

Combining it with the product law…


Combining it with the product law1

Combining it with the product law…

=


Always simplify using the power law first

Always simplify using the power law first!

=


Always simplify using the power law first1

Always simplify using the power law first!

=


Now use the product law

Now use the product law…

=


Now use the product law1

Now use the product law…

=


One more

One more…


One more1

One more…


Power law 1 st

Power Law 1st:


Power law 1 st1

Power Law 1st:


Power law 1 st2

Power Law 1st:


Your turn

Your turn…


Your turn1

Your turn…


Your turn2

Your turn…


Your turn3

Your turn…


Your turn4

Your turn…


Your turn5

Your turn…


Gentle reminder

Gentle reminder…


Gentle reminder1

Gentle reminder…

Coefficients are raised to the exponent


Example

Example:


Example1

Example:


Example2

Example:


Application

Application:


Application1

Application:

Raising a product to a power


Application2

Application:

Raising a product to a power


Examples12

Examples:


Examples13

Examples:


Examples14

Examples:


Examples15

Examples:


Examples16

Examples:


Examples17

Examples:


Go to town

Go to town:


Go to town1

Go to town:


Go to town2

Go to town:


Go to town3

Go to town:


Life lesson handle negatives at the end of the problem

Life lesson: handle negatives at the end of the problem.


Your turn again

Your turn again:


Your turn again1

Your turn again:

=


Your turn again2

Your turn again:

=


Your turn again3

Your turn again:

=


Your turn again4

Your turn again:

=


Scientific notation returns

Scientific Notation Returns…


Scientific notation returns1

Scientific Notation Returns…


Scientific notation returns2

Scientific Notation Returns…


Real world

Real world…


Real world1

Real world…

  • The expression gives the kinetic energy, in joules, of an object of mass of m kg traveling at a speed of v meters per second.


Real world2

Real world…

  • What is the kinetic energy of an experimental unmanned jet with a mass of kg traveling at a speed of about m/s?


Solution

Solution:


Solution1

Solution:

=


Solution2

Solution:

=

=


Solution3

Solution:

=

=

½ (


Solution4

Solution:

=

½ (

joules


Last law

Last Law…


Last law1

Last Law…

  • Moving in the opposite direction


Example3

Example:


Again with expanded notation

Again, with expanded notation:


Again with expanded notation1

Again, with expanded notation:


Again with expanded notation2

Again, with expanded notation:

What are a great deal of the 4’s going to do to each other?


Cancel out

Cancel Out!

What are a great deal of the 4’s going to do to each other?


Cancel out1

Cancel Out!

What are a great deal of the 4’s going to do to each other?


Cancel out2

Cancel Out!

What are we left with?


Cancel out3

Cancel Out!

What are we left with?


Cancel out4

Cancel Out!

What are we left with?

Which is equivalent to…


Cancel out5

Cancel Out!

What are we left with?

Which is equivalent to…


What s the magic math way of turn 5 and 3 into 2

What’s the “magic” math way of turn 5 and 3 into 2?

What are we left with?

Which is equivalent to…


Division

Division!!!

What are we left with?

Which is equivalent to…


Division property of exponents

Division Property of Exponents…


Division property of exponents1

Division Property of Exponents…


Division property of exponents2

Division Property of Exponents…

When dividing powers with the same base, we subtract their exponents.


Division property of exponents3

Division Property of Exponents…

ONE RESTRICTION:


Division property of exponents4

Division Property of Exponents…

ONE RESTRICTION: a  0


Does everyone know why

Does everyone know why?

ONE RESTRICTION: a  0


Quick hits

Quick hits


Quick hits1

Quick hits


Quick hits2

Quick hits


Quick hits3

Quick hits


Helpful hint saving time

Helpful Hint: Saving Time


Helpful hint saving time1

Helpful Hint: Saving Time

  • When the larger exponent is in the denominator,


Helpful hint saving time2

Helpful Hint: Saving Time

  • When the larger exponent is in the denominator,

    you can subtract

    the top from the

    bottom and put

    your answer in the denominator


Does anyone need to see an example

Does anyone need to see an example?

  • When the larger exponent is in the denominator,

    you can subtract

    the top from the

    bottom and put

    your answer in the denominator


Fractions

Fractions:


Fractions1

Fractions:


Fractions2

Fractions:


Multiple variables

Multiple Variables


Your try

Your try


Your try1

Your try


Your try2

Your try


Your try3

Your try


Your try4

Your try


Your try5

Your try


Scientific notation population density

Scientific Notation & Population Density


Scientific notation population density1

Scientific Notation & Population Density

Population Density is defined as:


Scientific notation population density2

Scientific Notation & Population Density

Population Density is defined as:


Example4

Example


Example5

Example

  • If the country of Wazup has a population of 3.5 • and a land size of 7 •,what is its population density?


Example6

Example

  • If the country of Wazup has a population of 3.5 • and a land size of 7 •,what is its population density?


Last application

Last Application


Last application1

Last Application

  • Raising an exponent to a power:


Last application2

Last Application

  • Raising an exponent to a power:


Last application3

Last Application

  • b  0


Last application4

Last Application

  • When raising a fraction to an exponent


Last application5

Last Application

  • When raising a fraction to an exponent

    raise both the numerator and denominator to the exponent.


Examples18

Examples:


Examples19

Examples:


Examples20

Examples:


Examples21

Examples:


Examples22

Examples:


Choices choices

Choices, choices…


Choices choices1

Choices, choices…

  • Simplify the following expression:


Suggested

Suggested:


Your choice

Your choice…


You try2

You try:


You try3

You try:


You try4

You try:


You try5

You try:


What would a lesson be without negative exponents

What would a lesson be without negative exponents?


What would a lesson be without negative exponents1

What would a lesson be without negative exponents?


What would a lesson be without negative exponents2

What would a lesson be without negative exponents?


Example7

Example:


Example8

Example:


Example9

Example:


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