Class Notes on The Exponent Rules



FOM2 - UNIT 7 – NOTE PACKET

FOM2 Unit 7 Notes: DAY 1 – Exponent Rules

Warm-up

Use trig to solve for x.

1. x = ______ 2. x = ______ y = ______

Day 1 Notes: The Exponent Rules

What is a POWER?

When it comes to mathematics, we refer to a _______________________ as an exponential expression used for indicating the numbers of factors involved in multiplication.

There are special rules for powers that will make solving questions in this course easier if we apply them correctly. You may have been exposed to these rules way back in the day....

What is a BASE?

What is an EXPONENT?

The Exponent Rules

Rule #1: Multiplying Powers with the Same Base

When multiplying powers with the same base you _________________ the exponents.

Example: a3 x a2 = a3 + 2 = a5

You try:

1.

2.

3.

Rule #2: Dividing Powers with the Same Base

When dividing powers with the same base you _____________________ the exponents.

Example: a5 [pic] a3 = a5 – 3 = a2

You try:

1.

2.

3.

Rule #3: When Raising Powers to Another Power

When raising a power to another power you __________________________ the exponents.

Example: (a4)2 = a4 x 2 = a8

You try:

1.

2.

3.

Rule #4: Powers with a Negative Exponent

Powers with a negative exponent can be written as a FRACTION with a _______________________________ exponent.

Example: a-5 = [pic]

You try:

1.

2.

3.

Rule #5: A Power with an Exponent of One

When evaluating a power with an exponent of one, the answer will be the base.

Example: a1 = a

You try:

1.

2.

3.

Rule #6: A Power with an Exponent of Zero

When evaluating a power with an exponent of zero, the answer will be one.

Example: a0 = 1

You try:

1.

2.

3.

Day 1 Classwork

Find the Value of Each Expression:

1) [pic] 2) [pic] 3) [pic] 4) [pic]

5) [pic] 6) [pic] 7) [pic]

Simplify Each Product:

8) [pic] 9) [pic] 10) [pic]

11) [pic] 12) [pic]

Simplify Each Product:

13) [pic] 14) [pic]

15) [pic] 16) [pic]

17) [pic]

Simplify Each Expression:

18) [pic] 19) [pic] 20) [pic]

21) [pic] 22) [pic] 23) [pic]

24) [pic] 25) [pic] 26) [pic]

27) [pic] 28) [pic]

Simplify each Quotient and then find the Value of the Result:

29) [pic] 30) [pic]

Simplify Each Expression:

31) [pic] 32) [pic] 33) [pic]

34) [pic] 35) [pic] 36) [pic]

FOM2 Unit 7 Notes: DAY 2 – Exponent Rules Practice

1. Set up MATHO Card #1

2. Get a white board marker and eraser

3. Get a highlighter

Game Board #1

|M |A |T |H |O |

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| | |FREE | | |

| | |SPACE | | |

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Game Board #2

|M |A |T |H |O |

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| | | | | |

| | |FREE | | |

| | |SPACE | | |

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FOM2 Unit 7 Notes: DAY 3 – Simplifying Radicals

1) [pic]

2) [pic]

3) [pic]

Day 3 Notes

____________________________ is the inverse function of exponents.

[pic]

An _____________________ in a radical tells you how many times you have to multiply the root times itself to get the radicand.

Ex) [pic]______________________________________

81 = ____________________________________________

9 = _____________________________________________

2 = _____________________________________________

When a radical is written without an index, there is an understood index of 2.

Examples:

[pic] = _______________________________________________

Radicand: ___________________ Index: ______________________

Root is ___________ because ____________________=________=_________

[pic] = __________________________________________

Radicand: ___________________ Index: ______________________

Root is ___________ because ____________________=________=_________

Yes….. you can use a calculator to do this, but for some of the more simple problems, you should be able to figure them out in your head.

To use the calculator:

1) Math

2) Select #5: [pic]

3) Type in index and radicand

4) Enter

Discuss with your neighbor if you’ve forgotten the shortcuts to find square roots and cube roots using your calculator.

You try:

1) [pic]

2) [pic]

3) [pic]

Not every problem will work out that nicely! Try using our calculator to find an exact answer for [pic].

The calculator will give us an estimation, but we CAN’T write down an irrational number like this exactly – it can’t be written as a fraction and the decimal never repeats or terminates. The best we can do for an exact answer is use simplest radical form.

Simplifying Radical Steps:

1) Factor Tree (Both Numbers and Variables)

2) Group according to the index

3) Simplify ( Negative numbers under radical are okay for odd index’s but create imaginary numbers( _________ ) for even index’s.

Ex) [pic]

Ex)[pic]

Ex) [pic]

Day 3 Practice

Simplify the Radicals completely. (Notice all have an index of 2).

[pic]

FOM2 Unit 7 Notes: DAY 4 – Simplifying Radicals Practice

Let’s review Simplifying Radicals with different indexes.

[pic]

FOM2 Unit 7 Notes: DAY 5 – Rational Exponents

Warm-up

Simplify the following:

1) [pic]

2) [pic]

Day 5 Notes

Simplifying Exponents:

|Evaluate each expression. |Where would [pic] and [pic] fall in this list? |

|[pic] | |

| |Enter [pic]into the calculator. |

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| |[pic] |

| |Enter [pic]into the calculator. |

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| |[pic] |

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|______________________________ Exponents are another way to write radicals. |

| |

|Raising a number to the power of ½ is the same as performing a _____________ root. |

|Bases with fraction exponents can be written as radicals |Radical expressions can be written with bases with fractions for exponents|

|[pic] |[pic] |

We have already done this when simplifying radicals:

[pic] and [pic]

Even if a fraction cannot reduce evenly, we can still write radicals using fraction exponents.

Write using rational (fraction) exponents:

[pic] [pic] [pic]

[pic] [pic]

Write each in Radical Form.

1.) [pic] 2.) [pic]

3.) [pic] 4.) [pic]

Day 5 Practice

You try: Rewrite each of the following expressions in radical form. Then simplify the radical if necessary.

|1. |2. |3. |4. |

|[pic] |[pic] |[pic] |[pic] |

|5. |6. |7. | |

|[pic] |[pic] |[pic] | |

Now, reverse the rule you developed to change radical expressions into rational expressions.

|1. |2. |3. |

|[pic] |[pic] |[pic] |

|4. |5. |6. |

|[pic] |[pic] |[pic] |

FOM2 Unit 7 Notes: DAY 6 – Solving Radical Equations

Warm-up

1. A ladder is leaning up against a house. The bottom of the ladder is 3 ft away from the building and the ladder makes an angle of 75 degrees with the ground.

a) How high up the building does the ladder reach?

b) How long is the ladder?

Day 6 Notes

Solving Radicals are very similar to the ___________root method in Unit 6.

Steps:

1) Get the base alone (aka. get the [pic] by itself!!!)

2) Take the inverse (the exponent should match the index)

3) Solve for the variable.

4) Check Answers for extraneous solutions.

***Special Cases:

- If index is EVEN then you must check both the positive and negative solutions.

- If you have an EVEN index and the equation is equal to a negative, then the answer is NO SOLUTION.

- Always check solutions for extraneous case (solution that doesn’t work).

1) [pic] 2) [pic]

3) [pic] 4) [pic]

5) [pic] 6) [pic]

7) [pic] 8) [pic]

9) [pic] 10) [pic]

FOM2 Unit 7 Notes: DAY 7 – Solving Radical Equations

Warm-up: Quiz Review

Directions:

Questions 1 – 8: Simplify.

Questions 9: Write in simplest radical form.

Questions 10: Write as a rational exponent.

|1. [pic] |2. [pic] |3. [pic] |4. [pic] |5. |

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|6. |7. |8. |9. |10. |

|[pic] |[pic] |[pic] |[pic] |[pic] |

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FOM2 Unit 7 Notes: DAY 8 – Graphing Radical Equations

Warm-up

Factor.

1. y = -16t2 – 48t

2. y = 5x2 – 10x

3. y = 52x2 – 13

4. y = x2 + 5x + 4

Day 8 Notes

Graphing Square Root Functions

Make a table for each function.

F(x) = x2 f(x) = [pic]

0 0

1 1

2 2

3 3

4 4

5 5

6 6

7 7

8 8

9 9

Ignore the points with decimals. What do you notice about the other points?

_________________________________________________________

These functions are _______________ of each other. By definition, this means the _____________ and the _____________ ______________.

Plot the points from the tables above.

As a result, the graphs have the same numbers in their points but the _____ and the _________ coordinates have ___________ _______________.

This causes the graphs to have the _____________ _______________ but to be __________________ over the line ____________.

The Square Root Function

Reflect the function f(x) = x2 over the line y = x.

Problems? _________________

We have to define the Square Root ______________ as _____________. This means that we will only use the ________________ side of the graph.

The result: f(x) = [pic] Characteristics of the graph

Vertex End Behavior

Domain

Range

Symmetry

Transforming the Graphs

Now that we know the shapes we can use what we know about transformations to put that shape on the coordinate plane.

Remember:

Translate Reflect Dilate

1) f(x) = [pic]

2) f(x) = [pic]

3) f(x) = [pic]

4) f(x) = [pic]

5) f(x) = [pic]

6) f(x) = [pic]

FOM2 Unit 8 Notes: DAY 9 – Radical Applications

Warm-up

Find any roots or zeros of each quadratic function below. Then explain what roots or zeros mean graphically.

1. f(x) = 2x2 – 5x – 3

2. f(x) = x2 + 2x + 1

3. f(x) = x2 + 2x + 3

4. f(x) = 2x2 + 3x - 1

Day 9 Notes

Steps:

1) Read problem fully.

2) Define terms.

3) Substitute values given into equation.

4) Solve for missing variable.

Examples:

1. A pendulum can be measured with the equation where T is the time in seconds, G is the force in gravity (10 m/s²) and L is the length of the pendulum.

a. Find the period if a pendulum is 0.9m long.

b. Find the period if the pendulum is 0.049m long.

c. How long would the pendulum be if the period were exactly 1 sec?

Use for #2 – 3.

2. If a plane flied at 30,000 feet, how far away is the horizon?

3. Janine was looking out across the ocean from her hotel room on the beach. Her eyes were 250 feet above the ground. She saw a ship on the horizon. Approximately how far was the ship from her?

4. When a car comes to a sudden stop, you can determine the skidding distance (in feet) for a given speed (in miles per hour) using the formula where s is skidding distance and x is speed. Calculate the skidding distance for the following speeds.

a. 55 mph

b. 65 mph

c. 75 mph

d. 40 mph

FOM2 Unit 7 Notes: DAY 10 – Variation

Warm-up

[pic]

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Day 10 Notes

Direct Variation: As x gets ___________________________________

A direct variation is represented by the equation ___________________ where k is the ________________________________.

1) Real world examples of direct variation include wages varying directly to hours worked or circumference of a circle varies directly as the diameter. Can you think of others?

2) If y varies directly as x and x=10 when y=9, then what is y when x=4?

3) The refund (r) you get varies directly as the number of cans (c) you recycle. If you get a $3.75 refund for 75 cans, how much should you receive for 500 cans?

Inverse Variation:_________________________________________.

Can be represented by the equation ____________________ where k is the constant of variation.

1) Real Word Examples of Inverse Variation – For a trip to Myrtle Beach, the greater your car speed, the ________________time it would take you to get there. Also, if a rectangle has an area of 15 square units, then as the length increases the width ________________________.

2) If y varies inversely as x and x=3 when y=9, then what is x when y=27?

3) The amount of resistance in an electrical circuit required to produce a given amount of power varies inversely with the square of the current. If a current of .8amps requires a resistance of 50 ohms, what resistance will be required by a current of .5 amps?

Day 10 Classwork

Find the Missing Variable:

1) y varies directly with x. If y = -4 when x = 2, find y when x = -6.

2) y varies inversely with x. If y = 40 when x = 16, find x when y = -5.

3) y varies inversely with x. If y = 7 when x = -4, find y when x = 5.

4) y varies directly with x. If y = 15 when x = -18, find y when x = 1.6.

Classify the following as: a) Direct b) Inverse c) Neither

5) m = -5p 6) c = [pic] 7) c = 3v + 1

8) r = [pic] 9) n = ½ f 10) u = [pic]

11) d = 4t - 2 12) z = [pic] 13) y = 5x + 10

What is the constant of variation for the following?

14) d = 4t 15) z = [pic] 16) n = ½ f 17) r = [pic]

Answer the following questions.

18) If x and y vary directly, as x decreases, what happens to the value of y?

19) If x and y vary inversely, as y increases, what happens to the value of x?

20) If x and y vary directly, as y increases, what happens to the value of x?

21) If x and y vary inversely, as x decreases, what happens to the value of y?

FOM2 Unit 7 Notes: DAY 11 – Variation Practice

Warm-up

1. Write each as a mixed radical, in simplest form:

A) [pic] B) [pic] C) [pic] D) [pic] E) [pic]

F) [pic] G) [pic] H) [pic] I) [pic] J) [pic]

-----------------------

y

8

x f(x)

x f(x)

[pic]

[pic]

[pic]

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