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Math Review
Pg. # 1 of 10
Math Review (8th)
1. Exponents:
Example 1.)
Example 2.)
Example 3.)
Example 4.)
Example 5.)
Example 6.)
Base
→32← Exponent
31 = 3 (any number to the 1st power equals itself)
32 = 3 • 3 = 9
33 = 3 • 3 • 3 = 27
30 = 1 (any number to the zero power equal 1)
3-1 = 1/3 (any number to a negative power equals 1 over the number to the
positive power)
-2
3 = 1/32 = 1/9
2. Square Root: One of the two equal factors of a number
 The √ symbol is called a radical sign.
 Perfect squares are the squares of whole numbers.
o Example: The numbers 4, 9, 16, 25, 36, 49, and 64 are examples of perfect
squares (This is because 22, 32, 42, 52, 62, 72, 82, etc.)
 Negative numbers cannot have a square root because there is no number that can be
multiplied by itself to produce a negative number. (A negative times a negative is a
positive.)
√9 =3,
√169 = 13,
√30.25 = 5.5,
√2 = 1.414213562
3. Absolute Value: always gives a positive value of the number
Example 1. ) │-3│ = 3 and │3│ = 3
Example 2.)
│5│ = 5 and │-5│ = 5
Example 3.)
│-9 + 3│ = │-6│ = 6
4. Additive Inverse:
Example 1.)
Example 2.)
Example 3.)
the opposite of a number
the additive inverse of -1 is 1
the additive inverse of ¼ is -¼
the additive inverse of 5 is -5
5. Multiplicative Inverse: (means reciprocal)
Example 1.) the multiplicative inverse of 5 (or 5/1) is 1/5
Example 2.) the multiplicative inverse of 3/8 is 8/3
Example 3.) the multiplicative inverse if -¼ is -4/1 or simply -4
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6. Sets of Numbers:
Natural Numbers – the counting numbers
{1, 2, 3, 4, 5, 6, 7, 8,…}
Whole Numbers – zero plus the counting numbers {0, 1, 2, 3, 4, 5, 6, 7, 8,…}
Integers – the positive and negative whole numbers {…, -4, -3, -2, -1, 0, 1, 2, 3, 4,…}
Rational Numbers - A number that can be put into fraction form.
Example. { ½,
5 = 5/1,
0,
-20}
A decimal that ends (terminates) or repeats in a pattern.
Example. { -4.1,
3.5,
4.1212,
.25,
√25 = 5}
Irrational Numbers - Numbers which never end and never repeat in a pattern.
Example. { √2 = 1.414213562… or π = 3.141592654…}
6. Roman Numerals:
I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, M = 1000
7. Prime or Composite:
Prime – a whole number greater than one whose only factors are 1 and itself.
The first seven prime numbers are {2, 3, 5, 7, 11, 13, 17}
Composite – a whole number greater than one that is not prime.
The first seven composite numbers are {4, 6, 8, 9, 10, 12, 14}
8. Greatest Common Factor (GCF): Greatest # that can fit into both #’s.
Example 1.) factors of 32: 1, 2, 4, 8, 16, 32
factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
The GCF of 32 and 24 is 8.
9. Least Common Multiple (LCM): Smallest # that all #’s have in common if you skip count.
Example 1.) multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75
Multiples of 12: 12, 24, 36, 48, 60, 72
The LCM of 5 and 12 is 60
10. Prime Factorization:
Example 1.)
Answer:
22 x 3 x 5
60
6
2
10
3
2
5 ← Factor until you get all primes
Math Review
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11. Properties of numbers:
Commutative Property:
a+b=b+a
or
a•b=b•a
(the order of addition or multiplication does not affect the answer)
Associative Property: (a + b) + c = a + (b + c)
or
(a • b) • c = a • (b • c)
(the grouping of addition or multiplication does not affect the answer)
Distributive Property: a(b + c) = ab + ac
or
a(b - c) = ab – ac
(individual multiplication by a group of items in a set of parentheses)
Additive Identity Property: a + 0 = 0 + a = a
(Any real number plus 0 is the original number)
Multiplicative Identity Property: a • 1 = 1 • a = a
(Any real number times 1 is the original number)
Additive Inverse Property: a + (-a) = -a + a = 0
(The sum of a real number and its opposite is zero)
Multiplicative Inverse Property: a • 1/a = 1/a • a = 1
(The product of a nonzero real number and its reciprocal is 1)
Multiplication Property of Zero: a • 0 = 0
(any number times zero equals zero)
12. Rules for Operations on Signed Numbers
Addition:
1. If the signs are the same – keep the sign and add the numbers.
2. If the signs are different – subtract the smaller number from the larger
number and keep the sign of the larger number.
Subtraction:
1. Add the opposite of the second number
(Change the sign of the number following the minus sign, then follow the
rules of addition.)
Multiplication and Division:
1. If multiplying or dividing two numbers of the same sign, the answer is
positive.
2. If multiplying or dividing two numbers of different signs, the answer is
negative.
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13. Geometry:
Angles:
Supplementary angels – 2 angles whose sum is 180°
Complementary angels – 2 angles whose sum is 90°
Acute angle – an angle which is greater than 0° but less than 90°
Right angle – a 90° angle
Obtuse angle – an angle which is greater than 90° but less than 180°
Straight angle – a 180° angle
Angles formed by parallel lines and a transversal
Vertical angles are congruent (2 and 3)
(1 and 4)
1 2
3 4
5
7
6
8
Alternate interior angles are congruent: (3 and 6) or (4 and 5)
Alternate exterior angles are congruent: (1 and 8) or (2 and 7)
Corresponding angles are congruent: (2 and 6) or (4 and 8)
(1 and 5) or (3 and 7)
Parallel lines: Lines that extend infinitely without intersecting {symbol ║}
Example: AB ║ CD
A
B
C
D
Perpendicular lines: Two lines that form 90°
angles to each other. {symbol ┴}
JK ┴ LM
K
L
M
J
Bisect: To cut into 2 equal pieces
Similar Figures: Same shape, different size.
(the sides can change, but the angles remain the same)
(the sides of similar triangles are always in proportion to each other.)
Similar Triangles (Not Congruent)
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Congruent: Same exact size and shape.
Congruent Trapezoids
14. Classifying Polygons:
Polygon – a closed figure formed by three or more line segments
Regular Polygons – a polygon whose sides and angles are all equal
Naming Polygons
Triangle – 3 sides
Pentagon – 5 sides
Heptagon – 7 sides
Nonagon – 9 sides
Quadrilateral – 4 sides
Hexagon – 6 sides
Octagon – 8 sides
Decagon – 10 sides
15. Triangles:
Triangle – a polygon with exactly 3 sides
The sum of the angles of a triangle equals 180°
A triangle cannot have more than one 90° angle
A triangle cannot have more than one obtuse angle
Classification of triangles by angles
Acute triangle: has three acute angles
Right triangle: has 1 right angle
Obtuse triangle: has 1 obtuse angle
Classification of triangles by sides
Scalene triangle: no congruent sides
Isosceles triangle: has 2 congruent sides
Equilateral triangle: has 3 congruent sides
16. Classifying Quadrilaterals
Quadrilateral: A polygon with 4 sides and 4 angles
Parallelogram Opposite sides are parallel
Opposite sides are congruent
Trapezoid
Exactly 1 pair of parallel sides
Rectangle
Opposite sides are parallel
Opposite sides are congruent
All 4 angles are right angles
Opposite sides are parallel
All 4 sides are congruent
Rhombus
Square
Opposite sides are parallel
All 4 sides are congruent
All 4 angles are right angles
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Quadrilateral
Parallelogram
Rectangle
Trapezoid
Rhombus
Square
17. Area and Perimeter Notes
Perimeter:
The total distance around the outside edge of a geometric figure.
To find the perimeter of any shape, add up the lengths of all of its
sides.
Area:
The total number of square units enclosed by a geometric figure.
Area Formulas:
Rectangle
A = bh
base x height
or
Square
A = bh
base x height
or
Parallelogram
A = bh
base x height
Rhombus
Triangle
A = lw
length x width
A = s2
side squared
A = bh
base x height
A = ½ bh
½ x base x height
or
A = bh
2
(base x height) ÷ 2
b1
Trapezoid
b2
a = (b1 + b2)h
2
(base 1 x base 2 x height) ÷ 2
Math Review
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18. Circles
Circumference (C) – the distance around a circle
Diameter (D) – the distance across a circle through its center
Radius (r) - the distance from the center to the edge of a circle
Pi (π) – the ratio of the circumference to the diameter
(the value of π is approximately 3.14 or 22/7)
Chord – a line segment with endpoints on the circle
Arc – a part of the circumference of a circle
Formulas for Area and Circumference of a circle:
Circumference (C)
C = πd
Remember: Cherry Pie’s Delicious
(Circumference equals pi x diameter)
Area (A)
A = πr2
Remember: Apple Pies aRe 2
(Area equals Pi x radius squared)
19. Coordinate Plane:
1. Coordinate System – a system used to precisely locate points on a geometric
plane. It is formed by two intersecting number lines.
2. x-axis – the horizontal number line
3. y-axis – the vertical number line
4. Origin – the place where the two axes intersect. (0,0)
5. Quadrants – the four sections of the coordinate plane formed by the two
intersecting axes.
Quadrant
II
Quadrant
III
Quadrant
I
Quadrant
IV
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6. Ordered Pair – any location on the coordinate plane can be represented by an
ordered pair of numbers called coordinates. The coordinates are always in the
order (x,y).
7. x-coordinate (abscissa) – The first number in an ordered pair
8. y-coordinate (ordinate) - The second number in an ordered pair
20. Reflections and Symmetry
Transformation: A movement of a geometric figure
Reflection:
The image is flipped (mirrored) over a line
Translation: The sliding of an image
Dilation:
The enlargement or reduction of an image
Rotation:
Turn
Reflection in the x-axis
Reflection in the y-axis
Translation: slide
Dilation (enlargement) {the other type of dilation is a reduction}
Lines of symmetry through a geometric figure
A
Vert.
B
Horiz.
C
Horiz.
D
Horiz.
E
F
Horiz.
None
X
Both
Z
None
Rotational Symmetry – if a figure can be rotated less than 360° and still look like the original
figure.
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21. Scientific Notation: A way of expressing a number as the product of a number that is at least
1 but less than 10 and a power of 10.
1.) 678,000
2.) .000543
3.) 4,805,000
=
=
=
6.78 x 105
5.43 x 10-4
4.805 x 106
4.) 90,302,000,000,000
5.) .00000002
6.) .000702
*** When writing a number in scientific notation:
The decimal point always goes after the first digit
The answer should always be in the format: →
The exponent is positive for numbers greater than 1
(ex. 324,000,000)
The exponent is negative for numbers less than 1.
(decimals ex. .0000243)
=
=
=
9.0302 x 1013
2.0x10-8
7.02 x 10-4
__.__ __ x 10#
22. Order of Operations:
1 2 3
4
Rules for Order of Operations (P E MD AS)
FIRST: Do all operations within parentheses.
SECOND: Do all powers or exponents.
THIRD: Working from left to right, do all multiplication and division.
FOURTH: Working from left to right, do all addition and subtraction.
23. Pythagorean Theorem: (used to find the missing side of a right triangle when 2 sides are
given)
a2 + b2 = c2
a and b are legs
c is always the hypotenuse
c
a
b
24. Trigonometry of a Right Triangle: (can be used to find a missing angle or side of a right
triangle)
SOH CAH TOA
SIN = opposite
hypotenuse
COS= adjacent
hypotenuse
TAN= opposite
adjacent
25. Percent Problems:
Part (is) = Percent (%)
Whole (of)
100
1.) 42 is 50 percent of what number?
2.) 3.75 is what percent of 75?
3.) What is 12 ½ percent of 88?
4.) 12 is 30% of what number?
5.) 19 is what percent of 95?
6.) What percent of the area of the square is shaded?
Math Review
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26. Estimation of percent
25% = ¼
50% = ½
20% = 1/5
40% = 2/5
33 1/3% = 1/3
66 2/3% = 2/3
75% = ¾
60% = 3/5
10% = 1/10
80% = 4/5
27. Central Tendency
Mean –
the average (add up items and divide by the # of items)
Median –
the middle number when arranged in size order. If there is an even
number of elements you must average the two middle elements.
Mode –
the item that occurs the most.
28. Proportions: An equation where the two ratios are equal.
Example 1.)
2 = 8 In a proportion the cross products are equal (2 x 12 = 8 x 3)
3
12
29. Graphing inequalities
An inequality is a mathematical sentence that contains the symbols {<, > ,≤, or ≥}
When graphing inequalities use the following notation:
● Solid Endpoint means included example {≤ or ≥}
○ Hollow Endpoint means not included example {< or >}
→ Arrow endpoints extend infinitely
Example 1.
x≤3
Example 2.
x>2
Example 3. -5 ≤ x < 2
←|----|---|---|---|---|---|---|---|---●---|---|---|→
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
←|----|---|---|---|---|---|---|---○---|---|---|---|→
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
←|---●---|---|---|---|---|---|---○---|---|---|---|→
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
30. Solving inequalities:
To solve an inequality follow the same procedures as used in solving equations, with the
following exception:
 When multiplying or dividing both sides of an inequality by a negative number,
reverse the direction of the inequality.
Example 1
Example 2
Example 3
3x + 2 > 20
-5x – 4 > 21
-7x ≤ 28
-2
-2
+4 +4
3x
3
x
> 18
3
> 6
-5x
-5
x
> 25
-5
< -5
-7x ≤ 28
-7 -7
x ≥ -4