1. Changing Bases

2. Base 10: example number 2120 2 1 2 0 ₁₀ Base 8: 4110₈ 4 1 1 0 ₈
10³ ² ¹ ⁰ ₁₀
10³∙2 + 10²∙1 + 10¹∙2 + 10⁰∙0 = 2120₁₀
Implied base 10
Base 8: 4110₈
8³ ² ¹ ⁰ ₈
8³∙4 + 8²∙1 + 8¹∙1 + 8⁰∙0 = 2120₁₀
Base 8

4. Hexadecimal Numbers
Hexadecimal numbers are interesting. There are 16 of them!
They look the same as the decimal numbers up to 9, but then there are the letters ("A',"B","C","D","E","F") in place of the decimal numbers 10 to 15.
So a single Hexadecimal digit can show 16 different values instead of the normal 10 like this:
Decimal:
Hexadecimal: A B C D E F

5. Problem Solving:
3, 2, 1, … lets go!

6. Express the base 4 number 321₄ as a base ten number.

7. Answer: 57

8. Add:
23₄ + 54₈ = _______₁₀
(Base 10 number)

9. Answer: 55

10. 123.11₄ - 15.23₆ = ______₁₀ (Base 10 number)
Subtract:
123.11₄ ₆ = ______₁₀ (Base 10 number)

11. Answer:
15 ⁴³⁄₄₈

12. Express the base 10 number 493 as a base two number.

13. Answer: ₂

14. Add:
₁₀ ₁₀ =
________₁₀
(Base 10 number)

15. Answer:

16. Add: =
________4
(Base 4 number)

17. Answer:
1214

18. Add: =
________10
(Base 10 number)

19. Answer:
1214

20. Factorials

21. Factorial symbol ! is a shorthand notation for a special type of multiplication.

22. N! is written as N∙(N-1)∙(N-2)∙(N-3)∙ ….. ∙1 Note: 0! = 1
Example: 5! = 5∙4∙3∙2∙1
= 120

23. Problem Solving:
3, 2, 1, … lets go!

24. Solve:
6! = _____

25. Answer:
720

26. Solve: 5! 3!

27. Answer: 20

28. Solve: 5!
3!2!

29. Answer: 10

30. Squares

31. Positive Exponents “Squared”: a² = a·a example: 3² = 3·3 = 9

32. 0²= ²= ²=144 1²= ²= ²=169 2²= ²= ²=225 3²= ²= ²=256 4²= ²= ²=400 5²= ²= ²=625

33. What is the sum of the first 9 perfect squares?

34. Answer: =
285

35. Shortcut:
Use this formula
n(n+1)(2n+1) 6

36. Shortcut:
Use this formula
9(9+1)(2∙9+1) 6
Answer: 285

37. Square Roots

38. 9.1 Evaluating Roots 1. Find square roots.
2. Decide whether a given root is rational, irrational, or not a real number.
3. Find decimal approximations for irrational square roots.
4. Use the Pythagorean formula.
5. Use the distance formula.
6. Find cube, fourth, and other roots.

39. 9.1.1: Find square roots.
When squaring a number, multiply the number by itself. To find the square root of a number, find a number that when multiplied by itself, results in the given number. The number a is called a square root of the number a 2.

40. Find square roots. (cont’d)
The positive or principal square root of a number is written with
the symbol
The symbol – is used for the negative square root of a number.
Radical Sign
Radicand
The symbol , is called a radical sign, always represents the
positive square root (except that ). The number inside the
radical sign is called the radicand, and the entire expression—radical
sign and radicand—is called a radical.

41. Find square roots. (cont’d)
The statement is incorrect. It says, in part, that a positive number equals a negative number.

42. EXAMPLE 1 Find all square roots of 64. Solution:
Finding All Square Roots of a Number
Find all square roots of 64.
Solution:

43. Finding Square Roots
EXAMPLE 2:
Find each square root.
Solution:

44. EXAMPLE 3: Find the square of each radical expression. Solution:
Squaring Radical Expressions
Find the square of each radical expression.
Solution:

45. 9.1.2: Deciding whether a given root is rational, irrational, or not a real number.
All numbers with square roots that are rational are called perfect squares.
Perfect Squares
Rational Square Roots 25
144
A number that is not a perfect square has a square root that is irrational. Many square roots of integers are irrational.
Not every number has a real number square root. The square of a real number can never be negative. Therefore, is not a real number.

46. EXAMPLE 4:
Identifying Types of Square Roots
Tell whether each square root is rational, irrational, or not a real number.
Solution:
Not all irrational numbers are square roots of integers. For example  (approx ) is a irrational number that is not an square root of an integer.

47. 9.1.3: Find decimal approximations for irrational square roots.
A calculator can be used to find a decimal approximation even if a number is irrational.
Estimating can also be used to find a decimal approximation for irrational square roots.

48. EXAMPLE 5:
Approximating Irrational Square Roots
Find a decimal approximation for each square root. Round answers to the nearest thousandth.
Solution:

49. 9.1.4: Use the Pythagorean formula.
Many applications of square roots require the use of the Pythagorean formula.
If c is the length of the hypotenuse of a right triangle, and a and b are the lengths of the two legs, then
Be careful not to make the common mistake thinking that
equals

50. What is a right triangle?
hypotenuse
leg
right angle
leg
It is a triangle which has an angle that is 90 degrees.
The two sides that make up the right angle are called legs.
The side opposite the right angle is the hypotenuse.

51. The Pythagorean Theorem
In a right triangle, if a and b are the measures of the legs and c is the hypotenuse, then
a2 + b2 = c2.
Note: The hypotenuse, c, is always the longest side.

52. The Pythagorean Theorem
“For any right triangle, the sum of the areas of the two small squares is equal to the area of the larger.”
a2 + b2 = c2

53. Proof

54. Find the length of the hypotenuse if 1. a = 12 and b = 16.
= c2
= c2
400 = c2
Take the square root of both sides.
20 = c

55. Find the length of the hypotenuse if 2. a = 5 and b = 7.
= c2
= c2
74 = c2
Take the square root of both sides.
8.60 = c

56. Find the length of the hypotenuse given a = 6 and b = 12
180
324
13.42 18

57. Find the length of the leg, to the. nearest hundredth, if 3
Find the length of the leg, to the nearest hundredth, if 3. a = 4 and c = 10.
42 + b2 = 102
16 + b2 = 100
Solve for b.
b2 =
b2 = 84
b = 9.17

58. Find the length of the leg, to the nearest hundredth, if 4
Find the length of the leg, to the nearest hundredth, if 4. c = 10 and b = 7.
a = 102
a = 100
Solve for a.
a2 =
a2 = 51
a = 7.14

59. Find the length of the missing side given a = 4 and c = 51 3
6.4 9

60. 5. The measures of three sides of a triangle are given below
5. The measures of three sides of a triangle are given below. Determine whether each triangle is a right triangle , 3, and 8
Which side is the biggest?
The square root of 73 (= 8.5)! This must be the hypotenuse (c).
Plug your information into the Pythagorean Theorem. It doesn’t matter which number is a or b.

61. Sides: , 3, and = ( ) 2
= 73
73 = 73
Since this is true, the triangle is a right triangle!! If it was not true, it would not be a right triangle.

62. Determine whether the triangle is a right triangle given the sides 6, 9, and
Yes No
Purple

63. EXAMPLE 6 Find the length of the unknown side in each right triangle.
Using the Pythagorean Formula
EXAMPLE 6
Find the length of the unknown side in each right triangle.
Give any decimal approximations to the nearest thousandth.
Solution: 11 8 ?

64. EXAMPLE 7 A rectangle has dimensions of 5 ft by 12 ft. Find the length
Using the Pythagorean Formula to Solve an Application
A rectangle has dimensions of 5 ft by 12 ft. Find the length
of its diagonal.
5 ft
12 ft
Solution:

65. 9.1.5: Use the distance formula.
The distance between the points and is

66. EXAMPLE 8 Find the distance between and . Solution:
Using the Distance Formula
EXAMPLE 8
Find the distance between and
Solution:

67. 9.1.6: Find cube, fourth, and other roots.
Finding the square root of a number is the inverse of squaring a number. In a similar way, there are inverses to finding the cube of a number or to finding the fourth or greater power of a number.
The nth root of a is written
Radical sign
Index
Radicand
In , the number n is the index or order of the radical.
It can be helpful to complete and keep a list to refer to of third and fourth powers from 1-10.

68. Finding Cube Roots
EXAMPLE 9
Find each cube root.
Solution:

69. EXAMPLE 10
Finding Other Roots
Find each root.
Solution:

70. 9.2 Evaluating Roots 1. Multiply square root radicals.
2. Simplify radicals by using the product rule.
3. Simplify radicals by using the quotient rule.
4. Simplify radicals involving variables.
5. Simplify other roots.

71. 9.2.1: Multiply square root radicals.
For nonnegative real numbers a and b,
and
That is, the product of two square roots is the square root of
the product, and the square root of a product is the product of the square roots.
It is important to note that the radicands not be negative numbers in the product rule. Also, in general,

72. EXAMPLE 1 Find each product. Assume that Solution:
Using the Product Rule to Multiply Radicals
Find each product. Assume that
Solution:

73. 9.2.2: Simplify radicals using the product rule.
A square root radical is simplified when no perfect square factor remains under the radical sign.
This can be accomplished by using the product rule:

74. EXAMPLE 2 Simplify each radical. Solution:
Using the Product Rule to Simplify Radicals
Simplify each radical.
Solution:

75. EXAMPLE 3 Solution: Find each product and simplify.
Multiplying and Simplifying Radicals
Find each product and simplify.
Solution:

76. 9.2.3: Simplify radicals by using the quotient rule.
The quotient rule for radicals is similar to the product
rule.

77. EXAMPLE 4 Solution: Simplify each radical.
Using the Quotient Rule to Simply Radicals
Simplify each radical.
Solution:

78. EXAMPLE 5 Simplify. Solution:
Using the Quotient Rule to Divide Radicals
Simplify.
Solution:

79. EXAMPLE 6 Simplify. Solution:
Using Both the Product and Quotient Rules
Simplify.
Solution:

80. 9.2.4: Simplify radicals involving variables.
Radicals can also involve variables.
The square root of a squared number is always nonnegative. The absolute value is used to express this.
The product and quotient rules apply when variables appear under the radical sign, as long as the variables represent only nonnegative real numbers

81. EXAMPLE 7
Simplifying Radicals Involving Variables
Simplify each radical. Assume that all variables represent positive real numbers.
Solution:

82. 9.2.5: Simplify other roots.
To simplify cube roots, look for factors that are perfect
cubes. A perfect cube is a number with a rational cube root.
For example, , and because 4 is a rational number, 64 is a perfect cube.
For all real number for which the indicated roots exist,

83. Simplifying Other Roots
EXAMPLE 8
Simplify each radical.
Solution:

84. Simplify other roots. (cont’d)
Other roots of radicals involving variables can also be simplified. To simplify cube roots with variables, use the fact that for any real number a,
This is true whether a is positive or negative.

85. EXAMPLE 9 Simplify each radical. Solution:
Simplifying Cube Roots Involving Variables
Simplify each radical.
Solution:

86. Adding and Subtracting Radicals
9.3
Adding and Subtracting Radicals
1. Add and subtract radicals.
2. Simplify radical sums and differences.
3. Simplify more complicated radical expressions.

87. 9.3.1: Add and subtract radicals.
We add or subtract radicals by using the distributive property. For example,
Radicands are different
Indexes are different
Only like radicals—those which are multiples of the same root of the same number—can be combined this way. The preceding example shows like radicals. By contrast, examples of unlike radicals are
Note that cannot be simplified.

88. EXAMPLE 1 Add or subtract, as indicated. Solution:
Adding and Subtracting Like Radicals
Add or subtract, as indicated.
Solution:
It cannot be added by the distributive property.

89. 9.3.2: Simplify radical sums and differences.
Sometimes, one or more radical expressions in a sum or difference must be simplified. Then, any like radicals that result can be added or subtracted.

90. EXAMPLE 2 Add or subtract, as indicated. Solution:
Adding and Subtracting Radicals That Must Be Simplified
Add or subtract, as indicated.
Solution:

91. 9.3.3: Simplify more complicated radical expressions.
When simplifying more complicated radical expressions, recall the rules for order of operations.
A sum or difference of radicals can be simplified only if the radicals are like radicals. Thus, cannot be simplified further.

92. Simplifying Radical Expressions
EXAMPLE 3A
Simplify each radical expression. Assume that all variables represent nonnegative real numbers.
Solution:

93. EXAMPLE 3B
Simplifying Radical Expressions (cont’d)
Simplify each radical expression. Assume that all variables represent nonnegative real numbers.
Solution: