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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

Laws of Exponents

7-2 through 7-4













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 law2
Giving us our 1st law:

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







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.














First steps into1
First steps into…

Scientific Notation!!!



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 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 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?










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 law3
Power Law:

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




























Gentle reminder1
Gentle reminder…

Coefficients are raised to the exponent






Application1
Application:

Raising a product to a power


Application2
Application:

Raising a product to a power






















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?





Solution3
Solution:

=

=

½ (


Solution4
Solution:

=

½ (

joules



Last law1
Last Law…

  • Moving in the opposite direction





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 exponents2
Division Property of Exponents…

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



Division property of exponents4
Division Property of Exponents…

ONE RESTRICTION: a  0


Does everyone know why
Does everyone know why?

ONE RESTRICTION: a  0







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













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:



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 application1
Last Application

  • Raising an exponent to a power:


Last application2
Last Application

  • Raising an exponent to a power:



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.








Choices choices1
Choices, choices…

  • Simplify the following expression:














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