Description
In This collection, we will go deep into math.
This collection will help all math and high school students.
Study Set Content:
Luttrell 2012
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Name: ______________________
Date: _____
5h – Polynomials and Variation
Polynomials are algebraic e
x
pressions that involve only the operations of addition, subtraction,
and multiplication of variables. E
x
amples of polynomials are 3, 3
x
, 3
x
+ 1, 3
x
2
+ 6
x
+1, and
x
3
+
x
. Polynomials are described by the number of terms in the e
x
pression. Special names are
used for one-term through three-termed polynomials: monomial, binomial, trinomial.
Polynomials are also described by the highest-degree term in the e
x
pression: constant (zero-
degree), linear (first-degree), quadratic, cubic, quartic, quintic, he
x
tic (se
x
tic), heptic (septic), any
higher degrees are labeled by the ordinal value.
Match the following polynomials. If it isn’t a polynomial, e
x
plain wy:
___ A. 3
x
2
+ 6
x
+1
1. Quadratic trinomial
___ B. 3
x
2
+ 8x
4
2. Quintic monomial
___ C.
3
4
x
+
3. Non-polynomial; since ________________
___ D. 3
x
5
4. Non-polynomial, since division by variable
___ E. 5
5. He
x
tic binomial
___ F. 2
3
x
5
+
x
4
+ 3
6. Quartic binomial
___ G.
x
6
+
x
4
7. Constant
___ H.
8
4
x
+
8. Quintic trinomial
Ever had such a conversation?
“This recipe calls for eggs and flour.”
“How much flour?”
“One egg is needed for every cup of flour.”
The first statement tells what is needed, a simple list of ingredients. The third statement gives
the proportion of each amount. E
x
pressing how variables relate in an equation is very similar.
In the following exercises, you’ll get a chance to symbolically express that relationship of
variables. In the first couple of examples, you see the use of k. This letter represent the
proportion; its value is of secondary importance to the actually relationship of the variables. As
noted in the dialogue, the most important was getting the ingredients and then the proportion.
9.
y
varies directly with
x
.
(answer:
y
= k
x
)
10.
y
varies indirectly with
x
. (
y
= k/
x)
11.
y
varies with the reciprocal of
x
.
12.
y
varies with the cube of
x
.
13.
y
varies with the square root of
x
.
14.
y
is a constant.
15.
y
varies linearly with
x
.
16.
y
varies inversely with
x
.
17. Solve: The number of pounds you weigh is directly proportional to the number of kilograms you are.
Sally steps onto a scale calibrated in kilograms and finds that she is 60 kilograms. Her bathroom scale
says 140 pounds. Write the equation expressing pounds in terms of kilograms. How much would she
weigh if she was 25 kilograms?
Luttrell 2012
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Name: ______________________
Date: _____
5i – Identifying Functional Relations
Identify which of the following are functions. Determine the domain and range of each relation.
1.
2.
3.
4. 5.
6.
7.
8.
9.
Luttrell 2012
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Chapter 5 Test
Name: ___________________________
Date: ____________
SHOW WORK
. A calculator is allowed on this test. Attach any scratch paper that’s used.
1. Solve for x:
A. 2x - 6 = 9
B. -3(2 - x) = 18
C.
1
8
3
4
x
=
2. A truck contains crude oil. The mass of the empty truck is 14,000 kg. The mass of one barrel
of oil is 180 kg. Let T kilograms represent the total mass of the truck and the oil. Let b represent
the number of barrels of oil. Write an equation relating T and b. How much would the total
mass be, if the truck has 13 barrels?
3. Solve for
h
:
A.
A = ½bh
B.
V = πr
2
h
4. Graph the solutions:
a. 2y < -6
b. (5, -3)
c. 10 >
y ≥ 5
5. Graph the solutions:
a. | x – 3 | < 5
b. | x + 1 | ≤ 6
c. | 3x – 2 | = 4
10
5
10
5
10
Luttrell 2012
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Chapter 6 Linear Equations
Luttrell 2012
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Name: ______________________
Date: _____
6a – Linear Equations
Polynomials with
x
and y to the first power and are e
x
pressed either as A
x
+ By = C or
y = m
x
+ b are linear equations. They are called linear equations because when graphically
represented, the solutions form the shape of a line. There is a pattern between the solutions that
makes it easy to distinguish linear equations. In the spaces below, fill in the missing pieces to
the pattern:
Equations: A.
B.
C.
y
= 2
x
- 1
y
= -3
x
+ 2
2
x
+ 3
y
= 5
(-3, -7)
(0, 2)
(-5, 5)
(0, -1)
(1, -1)
(1, 1)
(3, 5)
(2, - 4)
(7, -3)
(6, 11)
(3, -7)
(13, -7)
In equation A, as
x
increased by___,
y
values increased by___. In equation B,
x
increased by___
each time
y
decreased by___. In equation C, increasing
x
by___ made
y
decrease by___.
It is necessary to find the solutions to a linear equation. Some people make a t-chart, where they
choose values of
x
(or
y
) and find the corresponding value of
y
(or
x
).
1. Fill in the t-chart for
x
- 2
y
= 6.
x
-2
1
2
y
-3.5 -3
-2
2. Fill in the t-chart for the following data: St. Joseph’s ice arena has a $3 admittance per person.
Let
x
represent the number of people in a group and y be the admittance price.
x
1
3
5
y
21
42
3. Let
x
represent a bag of apples that are being sold at $1.99 a bag. If
y
represents cost, fill in
the chart below:
x
4
12
y
11.94
19.90
Luttrell 2012
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Name: ______________________
Date: _____
6b – Graphs of Linear Equations
Some of the solutions to 2
x
-
y
= 1 are (-2,-5), (0,-1), and (1, 1). Recognizing 2
x
-y = 1 as a linear
equation, the conclusion can be made that there is a pattern between the ordered pairs. Between
the first two pairs,
x
increases by 2. But between the last two pairs, it increases by 1. With such
inconsistency, it will give some difficulty for writing an equation when only given coordinates.
So to circumvent this problem, slope was defined. Slope is the ratio of the change between y-
values to the change between
x
-values. As any ratio can be reduced to simplest terms, the ratio
4/2 between the first two ordered pairs reduces to 2, which is the ratio between the last two pairs.
Algebraically, slope (
m
) is defined as
m
y
y
x
x
=
−
−
1
2
1
2
. Other e
x
pressions are
rise
run
y
x
=
∆
∆
.
Slope can be found by taking the ratio between two ordered pairs, or by looking at the equation
when it is in the form
y = mx + b
(slope-intercept form). Solving for
y
in the e
x
ample gives
y = 2
x
- 1. If
x
= 0, the equation gives y = -1, just like the -1 in the equation. The y-intercept (b)
is represented by the -1. Go ahead and plot the point on the graph. By evaluating more values of
x
, more ordered pairs are obtained: (1,1) and (2, 3). The same can be done by starting at the
y
-
intercept and moving right 1 and up 2 and make a dot. Notice the slope was 2 and there is a
x
-
coefficient of 2 in the equation. So graphing can be done by knowing your
m
’s and
b
’s.
Graph the following:
1. 2
x
+
y
= 3
2. 6
x
+ 2
y
= 4
3. -6
x
+ 3
y
= 9
4.
y
=
x
- 3
5.
y
= 5
x
- 2
6.
y
= (½)
x
- 3
7.
y
= -2
x
+ 1
8. -
x
-
y
= 3
9.
y
= 3
x
Luttrell 2012
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Name: ______________________
Date: _____
6c –Writing Linear Equations
A needed skill is being able to write a model to represent values on a graph. A
model
is an
equation that represents the data. It may be an appro
x
imation, especially when the values do not
form a true shape of a line, parabola, etc. There are four equations to represent a line, and each
are useful when depending on a set of questions or given data. These equations are: slope-
intercept (y = m
x
+ b), point-slope (y - y
1
= m(
x
-
x
1
)), standard form (A
x
+ By = C, where A, B,
and C are integers), and double-intercept form (
x
/a + y/b = 1).
To write the equation of a line, you need either two points or a point and the slope. Let’s follow
two methods below with the points (1,-1) and (3,5). First find slope:
5
1
3
1
6
2
3
− −
−
= =
(
)
.
Method 1:
Method 2:
y
= m
x
+ b
y
-
y
1
=
m
(
x
-
x
1
)
y
= 3
x
+ b
y
- 5 = 3(
x
- 3)
5 = 3(3) + b
y
- 5 = 3
x
- 9
5 = 9 + b
y
= 3
x
- 4
- 4 = b
y
= 3
x
- 4
1. Write the equation that has slope
2. Write the equation that has m = 3
of ⅔ and y-intercept of -7.
and passes through (0,-2).
3. Write the equation that passes
4. Write the equation that passes
through (0,2) and (3,1).
through (0,3) and (2,1).
5. Write the equation that passes
6. Write the equation that passes
through (2,1) and (4,-3).
through (1,2) and (10,2).
7. Write the equation that passes through (-6, 3) and (3,7).
Luttrell 2012
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Name: ______________________
Date: _____
6d – More on Writing & Graphing Linear Equations
Write the equation of the line that has the following characteristics.
1. Slope = -3, point: (0,2)
2
. Slope = ⅔, point: (4,5)
3. Points: (6, 8) and (1, 7)
4. Points: (-2, 3) and (3, 8)
5. Slope = ¾ and passes through (4, 3)
6. Slope = -2 and passes through (3,-1)
7. Passes through (2,-3) and (4,5)
8. Passes through (-1,-3) and (3,-3)
Graph the equations after simplifying it.
9. 6x = 36
10. 7x = 42
11. 5x = -15
12. 3y = 12
13. 4y = -8
14. y = 0
15. y = 3x - 3
16. 4x - 3y = 6
17. 3x - 5y = 10
Luttrell 2012
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Name: ______________________
Date: _____
6e –Parallel Lines
A set of lines may be either parallel or not. No fuzzy logic here! Parallel lines never intersect.
Looking at a graph isn’t a good indicator because the lines could be getting closer little by little.
So determine if the lines are parallel by comparing the slopes.
If the slopes are equal, then the
lines are parallel.
If the slopes and intercepts are equal, then the lines are the same!
E
x
ample A: Is 2
x
-
y
= 5 parallel to 4
x
- 2
y
= 1?
Solution: Writing the equations into slope-intercept form:
y
= 2
x
- 5 and
y
= 2
x
- ½ , it is
obvious both have slopes of 2.
Y
es, the
y
are parallel.
E
x
ample B: Find the linear equation parallel to 3
x
-
y
= 5 that is passes through (0,2).
Solution: First find the slope, which is 3. Then using point-slope form, fill in the point and
the slope:
y
- 2 = 3(
x
-0). Simplif
y
the equation to slope-intercept form:
y
= 3
x
+ 2.
This equation can also be written as -3
x
+
y
= 2 or 3
x
-
y
= -2.
E
x
ample C: Find the linear equation parallel to 2
x
- 3
y
= 1 that passes through (1,2).
Solution: Slope of the line is ⅔, plug into point-slope form:
y
-
2 = ⅔(
x
-1). Simplif
y
into
standard form b
y
getting rid of fraction - multipl
y
b
y
denominator of slope:
3
y
- 6 = 2(
x
-1). Bring
x
and
y
together on one side of equation: -2
x
+ 3
y
= 4.
This equation can also be written as 2
x
- 3
y
= - 4.
There is a pattern in the examples that make finding parallel lines easier. Be on the look-out!
1. Determine which of the following lines are parallel. Show
y
our work.
A. 3
x
- 2
y
= 5
B. 6
x
+ 9
y
= 1 C. 6
x
- 4
y
= 4 D. 9
x
+ 6
y
= 1
2. Write the linear equation that is parallel to
x
- 3
y
= 1 and passes through the point (4, 2).
3. Write the linear equation that is parallel to 5
x
- 3
y
= 1 and passes through the point (- 4, 2).
4. Write the linear equation in standard form that is parallel to 4
x
+ 9
y
= 3 and intersects (4,-2).
Luttrell 2012
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Name: ______________________
Date: _____
6f – Perpendicular Lines
Lines that form at right angles are said to be perpendicular or orthogonal. If one line has a slope
of m, then the other has the slope of -1/m. The product between the slopes of perpendicular lines
is alwa
y
s -1. Another wa
y
of e
x
pressing the slopes is to sa
y
the slopes are negative reciprocals of
each other.
E
x
ample 1: Determine if the line 2
x
-
y
= 5 is perpendicular to
x
+ 2
y
= 3.
Solution: The equations can be written as
y
= 2
x
- 5 and
y
= (-½)
x
+ 1.5.
Since 2(-½) = -1, the lines are perpendicular.
E
x
ample 2: Determine if the lines 2
x
- 3
y
= 5 and 6
x
+ 4
y
= 3 are perpendicular.
Solution: The equations in slope-intercept form are
y
= (
⅔)
x
-5/3 and
y
= (-6/4)
x
+ ¾.
Since -
6/4 × ⅔ = -1, the lines are perpendicular. Note: -6/4 = -3/2 which is the
negative reciprocal of ⅔.
E
x
ample 3: Write the linear equation perpendicular to 2
x
-
y
= 5 which passes through (- 4,2).
Solution: The given line has slope of 2, so the perpendicular line must have m = -½.
Substitute the point and slope into the point-slope form to get
y
- 2 = -½(
x
+ 4).
Simplif
y
into slope-intercept form,
y
= (-½)
x
, or standard form,
x
+ 2
y
= 0.
There is a pattern in the examples that make finding perpendicular lines easier. Look out!
1. Determine which of the following lines are perpendicular. Show
y
our work.
A. 3
x
- 2
y
= 5
B. 6
x
+ 9
y
= 1 C. 6
x
- 4
y
= 4 D. 9
x
+ 6
y
= 1
2. Which of the following is perpendicular to -3
x
-
y
= 4? Show work.
A.
x
+ 3
y
= 2
B. 9
x
+ 3
y
= 3
C. 3
x
-
y
= 3
D.
x
- 3
y
= 5
3. Write the linear equation perpendicular to 5
x
- 3
y
= 1 that passes through (4, 2).
4. Write the linear equation in standard form that is perpendicular to 4
x
- 3
y
= 7 at (1,-1).
Luttrell 2012
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Name: ______________________
Date: _____
6g – Linear Inequalities
The solution to a linear inequalit
y
is a set of points that make the inequalit
y
true. In the case of
the inequalit
y
, the solutions include points that lie on one side of the line. The points on the line,
the boundar
y
to half-planar solutions, are solutions onl
y
if the inequalit
y
includes an =. Use the
following e
x
ample as a guide.
E
x
ample: 6
x
+ 3
y
> 12
3
y
> 12 - 6
x
y
> 4 - 2
x
Plot some points on the line
y
= -2
x
+ 4.
Connect the dots with a dashed line since the line itself contain
no solution to the inequalit
y
. Now pick a point that does NOT
lie on the line. (0,0) is an eas
y
choice. Since 0 + 0 > 12 is
false, then the other side of the dashed line must contain the
solutions. Shade the solution side.
Graph the solutions.
1. -2
x
-
y
> 4
2. 3
x
+
y
> 5
3.
x
-
y
> 4
4. -2
x
+
y
≤4
5.
x
+ 2
y
≥ 5
6. 4
x
- 3
y
< 9
Luttrell 2012
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Name: ______________________
Date: _____
6h – Systems of Equations- Substitution
System is to equation as a paragraph is to a sentence. Since all the sentences in the paragraph
relate to the same idea, all the equations in a s
y
stem relate to the same variables. The solution to
a system is the ordered pair(s) that makes all the equations true.
One wa
y
of finding the solution is to graph each equation and see where the graphs intersect.
Unfortunately if the graphs intersect an
y
where but a lattice point, it is hard to determine the e
x
act
values of the ordered pair(s). Instead most mathematicians either do a substitution or an
elimination process.
E
x
ample: solve the s
y
stem of 2
x
+
y
= 3 and 3
x
- 5
y
= -2.
Step I: choose one variable in one equation to solve
y
= 3 - 2
x
Step II: Substitute the e
x
pression into other equation
3
x
- 5(3-2
x
) = -2
Step III: Simplif
y
and solve for remaining variable
3
x
- 15 + 10
x
= -2
13
x
- 15 = -2
13
x
= 13
x
= 1
Step IV: Substitute known variable into either equation
2(1) +
y
= 3
2 +
y
= 3
y
= 1
Step V: write answer
Ans: (1,1)
Solve. Sketch a graph to confirm results.
1.
x
+
y
= 3
2. 5
x
-
y
= 6
3.
y
= -
x
+ 1
x
-
y
= 2
3
x
- 2
y
= -2
y
= -3
x
+ 5
4.
y
= 3
x
+ 5
5.
x
- 4
y
= 5
6. 4
x
- 2
y
= 2
y
= 2
x
+ 2 3
x
- 4
y
= -1 -3
x
+ 2
y
= 1
Luttrell 2012
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Name: ______________________
Date: _____
6i – Systems of Equations - Elimination
Solving a s
y
stem with Substitution either means you are good with fractions or you use it mostly
when a variable has a coefficient of one. Most times elimination will prove easier.
E
x
ample: Solve the s
y
stem: 5
x
+ 3
y
= 18
3
x
- 2
y
= 7
Step I: choose variable
y
ou want to eliminate. We choose to eliminate
y
for this example.
Step II: Multipl
y
each row b
y
a number so that the coefficients for
y
are opposites. The first
equation was multiplied by 2 and the second by 3. The equations are still balanced!
10
x
+ 6
y
= 36
9
x
- 6
y
= 21
Step III: Add the equations together. One variable must disappear or else we made a mistake!
By adding equal items to both sides, the result is a balanced equation.
19
x
+ 0 = 57
Step IV: Solve for remaining variable.
19
x
= 57
x
= 3
Step V: Repeat process with other variable or do a substitution.
3(3) - 2
y
= 7
9 - 2
y
= 7
-2
y
= -2
y
= 1
Solve. Sketch a graph to confirm results.
1.
x
+
y
= 5
2. 4
x
+ 3
y
= 7
3.
y
= -2
x
+ 1
x
-
y
= 1
3
x
- 2
y
= 1
y
= -3
x
+ 3
4. 5
x
-
y
= 6
5. 5
x
- 3
y
= 12
6. 4
x
+ 5
y
= 2
3
x
- 2
y
= -2 3
x
+ 2
y
= 11 3
x
+ 2
y
= 5
Luttrell 2012
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Name: ______________________
Date: _____
6j – Systems of Equations – Mixed Review
Solve the following systems algebraically. Decide whether to use substitution or elimination. But you
cannot
do only one method for the entire worksheet!!! Graph the system and confirm results.
1. 2x + 3y = 2
2. 5x+2y = 11
3. y = 2x - 1
4x - 9y = -1 x + y = 4 y = 3x - 5
4. x - y = -11
5. 6x - 7y = 47
7x + 4y = -22
2x + 5y = -21
Complete the sentence!
6. Two lines intersect zero, _________, or infinite times. The systems that contain these lines are
referred to as inconsistent (formed by parallel lines), independent (lines intersecting), or dependent (lines
that are the same).
Luttrell 2012
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Name: ______________________
Date: _____
6k – Linear Word Problems
You can identif
y
a linear equation even if it is written in the jargon of a word problem. The
biggest clue is to identif
y
a constant change in the values. If you read the cost for a pencil is
$0.50, then two pencils should cost $1.00. Then identify
y
our variables, choosing those not
easily mistaken for a number. Convert given values into ordered pairs. Find the slope and then
write the equation. Once
y
ou have an equation, you can predict almost an
y
thing.
Solve:
1. Pencils are sold at the bookstore for 49 cents each. How much would 75 pencils cost?
2. Some reception halls have a flat fee for use of the hall and then a fee of $25 or more for each
person the
y
will be serving. For a wedding reception, Black Forest Inn charges about $64 per
person plus $5500 for the use of their inn. How much will it cost my brother to host a wedding
of 250 people?
3. An auto repair shop charges $25 an hour. They say your muffler needs to be replaced for
$65. It takes them 2.5 hours to fi
x
your e
x
haust s
y
stem. How much will the bill be?
4. To frame an oil painting, framers charge a rate based on the perimeter of the painting.
You choose a polished wooden frame that has a price tag of $8 a foot. Will the price
of framing e
x
ceed
y
our $125 budget if
y
our painting is 3.5' b
y
2.5'?
5. A Nissan Sentra sold in 1996 for $17,000. Ten years later its worth is $3,400. How much
would it be worth in another five years, if the depreciation followed a linear model? (In real life,
cars depreciate exponentially.)
Luttrell 2012
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Name: ______________________
Date: _____
6L – More Word Problems (Mixed Review)
Solve the word problems:
1. When water freezes, it is 0
̊
C (or 32
̊
F). When water reaches its boiling point, it is 100
̊
C (or 212
̊
F).
What is the temperature in Fahrenheit if the water is 72
̊
C?
2. Machinery is purchased for $450, but five years later is worth $0. What would be the worth of the
machinery after two years? (Assume linear depreciation.)
3. A room is twice as long as the width. If the area of the room is 72 square feet, what is the perimeter?
4. A lawn is 1000 feet around. If the length is three times the width, how much square feet is the lawn?
5
. A recipe wants 4 cups of flour and 1 cup of oats. How much flour is needed for ⅔ cups of oats?
6. A recipe wants 3 cups of flour and ½ cup of butter. How much butter would you use for 8 cups of
flour?
7. Keith is 16 years older than Shirleen and three times as old as Rachel. If the sum of their ages is 96,
how old are they?
Luttrell 2012
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Chapter 6 Test
Name: ___________________________
Date: ____________
SHOW WORK
. A calculator is allowed on this test. Attach any scratch paper that’s used.
1. Solve for the variable:
A. -3(n + 2) = 21 B.
3
4
3
8
p
=
C.
1
3
5
− =
x
2. An airplane travels eight times as fast as a car. The difference in their speeds is 420 km/h.
How fast is each vehicle travelling?
3. Graph the following lines. Label intercepts.
a. y = -3
B. x = 4
C. y = 2x - 3
4. Graph the solutions:
a. 2y < -6
b. y < 2x - 3
c. 2x - y
≥5
5. Solve the system:
3x - 2y = 6
x + 2y = 10
5
5
10
10
5
Luttrell 2012
98
Chapter 6 Test, continued
6. Rewrite each equation into standard form (Ax + By = C) using integer coefficients.
A.
y
x
=
−
2
3
8
9
B. y - 3 = ½ (x + 4)
7. Write the linear equation that passes through (4, 3) and (3, 9). Show work for slope.
8. Plot the ordered pairs and label: ( 2, -3), (-4, 5) and (0, 6).
9. Write the linear equation that passes through (3, 5) and (7, 8).
10. W
rite the equation of the line parallel to 2x – 3y = 5 and passes through the point (4,-3). For three
bonus points, find the equation of the line perpendicular to 2x – 3y = 5 that passes through (4,-3).
Bonus: (3 pts) If x + y = 5 and x - y = 1, what is the value of
2
x^2 – y^2
?
(3 pts) If 2010x + 2010y = 2011(x + y), what is the value of x/y?
5
5
5
5
5
0
Luttrell 2012
99
Chapter 7 Exponents
Prior Skills:
•
Sheet 7a: division by zero
Luttrell 2012
100
7 - Rules of Exponents
The operation of raising a number to a power is e
x
ponentiation. In the e
x
pression
x
3
, 3
x
’s are
being multiplied together so that
x
3
=
x
∙
x
∙
x
. For the e
x
pression
x
n
,
x
is called the base, n is the
e
x
ponent and the whole e
x
pression is a power.
Remember that e
x
ponents occur second in the order of operations. So that 4
x
3
does not mean
4
x
×4
x
×4
x
, rather 4×
x
×
x
×
x
. One of the common mistakes when simplif
y
ing -2
4
is to wrongl
y
use
the negative. The expression -2
4
means the negative of 2 to the fourth or the negative of 16
which is -16. If it is -2 that is raised to the fourth power, then it needs to be written as (-2)
4
.
The properties of E
x
ponents:
1. Product of two powers with equal bases:
x
x
x
a
b
a b
⋅
=
+
2. Quotient of two powers with equal bases:
x
x
x
a
b
a b
=
−
3. Power of a power:
(
)
x
x
a
b
ab
=
4. Power of a product:
(
)
xy
x y
a
a
a
=
5. Power of a quotient:
( )
x
y
a
a
a
x
y
=
6.
x
0 = 1, as long as
x
≠0.
7.
x
x
a
a
−
=
1
8.
x
x
a
b
a
b
=
Reminders:
1. Never confuse distribution of e
x
ponents - ONL
Y
distribute over multiplication and division,
never over addition or subtraction. e.g. (x
2
+y
3
)
2
≠ x
4
+y
6
but (x
2
y
3
)
2
= x
4
y
6
.
2. The product of two fractions is made from the product of the numerators over the product of
the denominators.
e.g. (2∕3) (5∕6) = 10∕18
E
x
ample a:
x y
y
x y
2
3
5
2
8
•
=
E
x
ample b:
x
y
xy
xy
x x y
y xy
x y
xy
x
y
x y
x
y
2
3
2
3
2
2
2
3
3
4
2
6
4 1
2 6
3
4
3
4
•
=
=
=
=
=
−
−
−
(
)
