Laws of exponents
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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

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

    • All told, how many 3’s?


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

    • =3 3 3 3 33

    • All told, how many 3’s?


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

    • =

    • All told, how many 3’s?


  • Is there a quicker way?

    • =

    • All told, how many 3’s?


  • Is there a quicker way?

    • =

    • All told, how many 3’s?

      Absolutely; we can gain the same answer by


  • Is there a quicker way?

    • =

    • All told, how many 3’s?

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


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 1st law:


Giving us our 1st law:

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


Examples:


Examples:


Examples:


Examples:


Make sure to differentiate between…


Make sure to differentiate between…

Exponents and Coefficients


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…

4(9) = 36


Example of what I mean…

4(9) = 36


Example of what I mean…

4(9) = 36


Example of what I mean…

4(9) = 36


Example of what I mean…

=


Example of what I mean…

=


Example of what I mean…

=

6(9) = 54


Example of what I mean…

=

6(9) = 54


You try…


You try…


First steps into…


First steps into…

Scientific Notation!!!


First steps into…


First steps into…

1 < |a| < 10


First steps into…

1 < |a| < 10

n is an integer.


Examples:

256,000

0.0041


Examples:

256,000

0.0041


Examples:

256,000

0.0041


Keep in mind:


Keep in mind:

  • a is negative when the original number is negative.


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:

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


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 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

O =


Solution: Dimensional Analysis

O =


Solution: Dimensional Analysis

O =


Solution: Dimensional Analysis

O =


Solution: Dimensional Analysis

O =


Solution: Dimensional Analysis

O =


Solution: Dimensional Analysis

O =


Next Law


Next Law

POWER LAW!!!


Next Law

POWER LAW!!!

Raising an exponent inside parentheses to another exponent.


Next Law

POWER LAW!!!

Example:


Next Law

POWER LAW!!!

Example:


Expanded!

POWER LAW!!!

Example:


Expanded!

POWER LAW!!!

Example: =


Expanded!

Return to the product law

Example: =


Expanded!

Return to the product law

Example: =


Expanded!

Which I can obtain much faster by…

Example: =


Multiplication

Which I can obtain much faster by…

Example: =


Power Law:


Power Law:


Power Law:


Power Law:

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


Examples:


Examples:


Examples:


Examples:


Examples:


One problem:


One problem:


One problem:


Combining it with the product law…


Combining it with the product law…

=


Always simplify using the power law first!

=


Always simplify using the power law first!

=


Now use the product law…

=


Now use the product law…

=


One more…


One more…


Power Law 1st:


Power Law 1st:


Power Law 1st:


Your turn…


Your turn…


Your turn…


Your turn…


Your turn…


Your turn…


Gentle reminder…


Gentle reminder…

Coefficients are raised to the exponent


Example:


Example:


Example:


Application:


Application:

Raising a product to a power


Application:

Raising a product to a power


Examples:


Examples:


Examples:


Examples:


Examples:


Examples:


Go to town:


Go to town:


Go to town:


Go to town:


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


Your turn again:


Your turn again:

=


Your turn again:

=


Your turn again:

=


Your turn again:

=


Scientific Notation Returns…


Scientific Notation Returns…


Scientific Notation Returns…


Real world…


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 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:

=


Solution:

=

=


Solution:

=

=

½ (


Solution:

=

½ (

joules


Last Law…


Last Law…

  • Moving in the opposite direction


Example:


Again, with expanded notation:


Again, with expanded notation:


Again, with expanded notation:

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


Cancel Out!

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


Cancel Out!

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


Cancel Out!

What are we left with?


Cancel Out!

What are we left with?


Cancel Out!

What are we left with?

Which is equivalent to…


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 are we left with?

Which is equivalent to…


Division!!!

What are we left with?

Which is equivalent to…


Division Property of Exponents…


Division Property of Exponents…


Division Property of Exponents…

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


Division Property of Exponents…

ONE RESTRICTION:


Division Property of Exponents…

ONE RESTRICTION: a  0


Does everyone know why?

ONE RESTRICTION: a  0


Quick hits


Quick hits


Quick hits


Quick hits


Helpful Hint: Saving Time


Helpful Hint: Saving Time

  • When the larger exponent is in the denominator,


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?

  • 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:


Fractions:


Multiple Variables


Your try


Your try


Your try


Your try


Your try


Your try


Scientific Notation & Population Density


Scientific Notation & Population Density

Population Density is defined as:


Scientific Notation & Population Density

Population Density is defined as:


Example


Example

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


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

  • Raising an exponent to a power:


Last Application

  • Raising an exponent to a power:


Last Application

  • b  0


Last Application

  • When raising a fraction to an exponent


Last Application

  • When raising a fraction to an exponent

    raise both the numerator and denominator to the exponent.


Examples:


Examples:


Examples:


Examples:


Examples:


Choices, choices…


Choices, choices…

  • Simplify the following expression:


Suggested:


Your choice…


You try:


You try:


You try:


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 exponents?


Example:


Example:


Example:


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