The document discusses square roots and radicals. It defines the square root operation as extracting the number that, when squared, equals the number inside the radical. It provides a table of common square numbers and their square roots that should be memorized. It also describes how to estimate the square root of numbers between known square numbers using the table as a reference.

1. Radicals and Pythagorean Theorem

2. Radicals and Pythagorean Theorem
From here on we need a scientific calculator.
You may use the digital calculators on your personal devices.

3. Radicals and Pythagorean Theorem
From here on we need a scientific calculator.
You may use the digital calculators on your personal devices.
Make sure the calculator could displace your input so that
you may check the input before executing it.

4. Radicals and Pythagorean Theorem
Square Root
From here on we need a scientific calculator.
You may use the digital calculators on your personal devices.
Make sure the calculator could displace your input so that
you may check the input before executing it.

5. “9 is the square of 3” may be rephrased backwards as
“3 is the square root of 9”.
Radicals and Pythagorean Theorem
Square Root
From here on we need a scientific calculator.
You may use the digital calculators on your personal devices.
Make sure the calculator could displace your input so that
you may check the input before executing it.

6. “9 is the square of 3” may be rephrased backwards as
“3 is the square root of 9”.
Radicals and Pythagorean Theorem
Definition: If a is > 0, and a2 = x, then we say a is the square
root of x. This is written as sqrt(x) = a, or x = a.
Square Root
From here on we need a scientific calculator.
You may use the digital calculators on your personal devices.
Make sure the calculator could displace your input so that
you may check the input before executing it.

7. “9 is the square of 3” may be rephrased backwards as
“3 is the square root of 9”.
Example A.
a. Sqrt(16) =
c.–3 =
Radicals and Pythagorean Theorem
Definition: If a is > 0, and a2 = x, then we say a is the square
root of x. This is written as sqrt(x) = a, or x = a.
b. 1/9 =
d. 3 =
Square Root
From here on we need a scientific calculator.
You may use the digital calculators on your personal devices.
Make sure the calculator could displace your input so that
you may check the input before executing it.

8. “9 is the square of 3” may be rephrased backwards as
“3 is the square root of 9”.
Example A.
a. Sqrt(16) = 4
c.–3 =
Radicals and Pythagorean Theorem
Definition: If a is > 0, and a2 = x, then we say a is the square
root of x. This is written as sqrt(x) = a, or x = a.
b. 1/9 =
d. 3 =
Square Root
From here on we need a scientific calculator.
You may use the digital calculators on your personal devices.
Make sure the calculator could displace your input so that
you may check the input before executing it.

9. “9 is the square of 3” may be rephrased backwards as
“3 is the square root of 9”.
Example A.
a. Sqrt(16) = 4
c.–3 =
Radicals and Pythagorean Theorem
Definition: If a is > 0, and a2 = x, then we say a is the square
root of x. This is written as sqrt(x) = a, or x = a.
b. 1/9 = 1/3
d. 3 =
Square Root
From here on we need a scientific calculator.
You may use the digital calculators on your personal devices.
Make sure the calculator could displace your input so that
you may check the input before executing it.

10. “9 is the square of 3” may be rephrased backwards as
“3 is the square root of 9”.
Example A.
a. Sqrt(16) = 4
c.–3 = doesn’t exist
Radicals and Pythagorean Theorem
Definition: If a is > 0, and a2 = x, then we say a is the square
root of x. This is written as sqrt(x) = a, or x = a.
b. 1/9 = 1/3
d. 3 =
Square Root
From here on we need a scientific calculator.
You may use the digital calculators on your personal devices.
Make sure the calculator could displace your input so that
you may check the input before executing it.

11. “9 is the square of 3” may be rephrased backwards as
“3 is the square root of 9”.
Example A.
a. Sqrt(16) = 4
c.–3 = doesn’t exist
Radicals and Pythagorean Theorem
Definition: If a is > 0, and a2 = x, then we say a is the square
root of x. This is written as sqrt(x) = a, or x = a.
b. 1/9 = 1/3
d. 3 = 1.732.. (calculator)
Square Root
From here on we need a scientific calculator.
You may use the digital calculators on your personal devices.
Make sure the calculator could displace your input so that
you may check the input before executing it.

12. “9 is the square of 3” may be rephrased backwards as
“3 is the square root of 9”.
Example A.
a. Sqrt(16) = 4
c.–3 = doesn’t exist
Radicals and Pythagorean Theorem
Definition: If a is > 0, and a2 = x, then we say a is the square
root of x. This is written as sqrt(x) = a, or x = a.
b. 1/9 = 1/3
d. 3 = 1.732.. (calculator)
Note that the square of both +3 and –3 is 9,
but we designate sqrt(9) or 9 to be +3.
Square Root
From here on we need a scientific calculator.
You may use the digital calculators on your personal devices.
Make sure the calculator could displace your input so that
you may check the input before executing it.

13. “9 is the square of 3” may be rephrased backwards as
“3 is the square root of 9”.
Example A.
a. Sqrt(16) = 4
c.–3 = doesn’t exist
Radicals and Pythagorean Theorem
Definition: If a is > 0, and a2 = x, then we say a is the square
root of x. This is written as sqrt(x) = a, or x = a.
b. 1/9 = 1/3
d. 3 = 1.732.. (calculator)
Note that the square of both +3 and –3 is 9,
but we designate sqrt(9) or 9 to be +3.
We say “–3” is the “negative of the square root of 9”.
From here on we need a scientific calculator.
You may use the digital calculators on your personal devices.
Make sure the calculator could displace your input so that
you may check the input before executing it.
Square Root

14. 0 02 = 0 0 = 0
1 12 = 1 1 = 1
2 22 = 4 4 = 2
3 32 = 9 9 = 3
4 42 = 16 16 = 4
5 52 = 25 25 = 5
6 62 = 36 36 = 6
7 72 = 49 49 = 7
8 82 = 64 64 = 8
9 92 = 81 81 = 9
10 102 = 100 100 = 10
11 112 = 121 121 = 11
Radicals and Pythagorean Theorem
Following are the square numbers and square-roots that one
needs to memorize.

15. 0 02 = 0 0 = 0
1 12 = 1 1 = 1
2 22 = 4 4 = 2
3 32 = 9 9 = 3
4 42 = 16 16 = 4
5 52 = 25 25 = 5
6 62 = 36 36 = 6
7 72 = 49 49 = 7
8 82 = 64 64 = 8
9 92 = 81 81 = 9
10 102 = 100 100 = 10
11 112 = 121 121 = 11
Radicals and Pythagorean Theorem
Following are the square numbers and square-roots that one
needs to memorize. These numbers are special because
many mathematics exercises utilize square numbers.

16. 0 02 = 0 0 = 0
1 12 = 1 1 = 1
2 22 = 4 4 = 2
3 32 = 9 9 = 3
4 42 = 16 16 = 4
5 52 = 25 25 = 5
6 62 = 36 36 = 6
7 72 = 49 49 = 7
8 82 = 64 64 = 8
9 92 = 81 81 = 9
10 102 = 100 100 = 10
11 112 = 121 121 = 11
We may estimate the sqrt
of other small numbers using
this table.
Radicals and Pythagorean Theorem
Following are the square numbers and square-roots that one
needs to memorize. These numbers are special because
many mathematics exercises utilize square numbers.

17. 0 02 = 0 0 = 0
1 12 = 1 1 = 1
2 22 = 4 4 = 2
3 32 = 9 9 = 3
4 42 = 16 16 = 4
5 52 = 25 25 = 5
6 62 = 36 36 = 6
7 72 = 49 49 = 7
8 82 = 64 64 = 8
9 92 = 81 81 = 9
10 102 = 100 100 = 10
11 112 = 121 121 = 11
We may estimate the sqrt
of other small numbers using
this table. For example,
25 < 30 < 36
Radicals and Pythagorean Theorem
Following are the square numbers and square-roots that one
needs to memorize. These numbers are special because
many mathematics exercises utilize square numbers.

18. 0 02 = 0 0 = 0
1 12 = 1 1 = 1
2 22 = 4 4 = 2
3 32 = 9 9 = 3
4 42 = 16 16 = 4
5 52 = 25 25 = 5
6 62 = 36 36 = 6
7 72 = 49 49 = 7
8 82 = 64 64 = 8
9 92 = 81 81 = 9
10 102 = 100 100 = 10
11 112 = 121 121 = 11
We may estimate the sqrt
of other small numbers using
this table. For example,
25 < 30 < 36
hence
25 < 30 <36
Radicals and Pythagorean Theorem
Following are the square numbers and square-roots that one
needs to memorize. These numbers are special because
many mathematics exercises utilize square numbers.

19. 0 02 = 0 0 = 0
1 12 = 1 1 = 1
2 22 = 4 4 = 2
3 32 = 9 9 = 3
4 42 = 16 16 = 4
5 52 = 25 25 = 5
6 62 = 36 36 = 6
7 72 = 49 49 = 7
8 82 = 64 64 = 8
9 92 = 81 81 = 9
10 102 = 100 100 = 10
11 112 = 121 121 = 11
We may estimate the sqrt
of other small numbers using
this table. For example,
25 < 30 < 36
hence
25 < 30 <36
or 5 < 30 < 6
Radicals and Pythagorean Theorem
Following are the square numbers and square-roots that one
needs to memorize. These numbers are special because
many mathematics exercises utilize square numbers.

20. 0 02 = 0 0 = 0
1 12 = 1 1 = 1
2 22 = 4 4 = 2
3 32 = 9 9 = 3
4 42 = 16 16 = 4
5 52 = 25 25 = 5
6 62 = 36 36 = 6
7 72 = 49 49 = 7
8 82 = 64 64 = 8
9 92 = 81 81 = 9
10 102 = 100 100 = 10
11 112 = 121 121 = 11
We may estimate the sqrt
of other small numbers using
this table. For example,
25 < 30 < 36
hence
25 < 30 <36
or 5 < 30 < 6
Since 30 is about half way
between 25 and 36,
Radicals and Pythagorean Theorem
Following are the square numbers and square-roots that one
needs to memorize. These numbers are special because
many mathematics exercises utilize square numbers.

21. 0 02 = 0 0 = 0
1 12 = 1 1 = 1
2 22 = 4 4 = 2
3 32 = 9 9 = 3
4 42 = 16 16 = 4
5 52 = 25 25 = 5
6 62 = 36 36 = 6
7 72 = 49 49 = 7
8 82 = 64 64 = 8
9 92 = 81 81 = 9
10 102 = 100 100 = 10
11 112 = 121 121 = 11
We may estimate the sqrt
of other small numbers using
this table. For example,
25 < 30 < 36
hence
25 < 30 <36
or 5 < 30 < 6
Since 30 is about half way
between 25 and 36,
so we estimate that30  5.5.
Radicals and Pythagorean Theorem
Following are the square numbers and square-roots that one
needs to memorize. These numbers are special because
many mathematics exercises utilize square numbers.

22. 0 02 = 0 0 = 0
1 12 = 1 1 = 1
2 22 = 4 4 = 2
3 32 = 9 9 = 3
4 42 = 16 16 = 4
5 52 = 25 25 = 5
6 62 = 36 36 = 6
7 72 = 49 49 = 7
8 82 = 64 64 = 8
9 92 = 81 81 = 9
10 102 = 100 100 = 10
11 112 = 121 121 = 11
We may estimate the sqrt
of other small numbers using
this table. For example,
25 < 30 < 36
hence
25 < 30 <36
or 5 < 30 < 6
Since 30 is about half way
between 25 and 36,
so we estimate that30  5.5.
In fact 30  5.47722….
Radicals and Pythagorean Theorem
Following are the square numbers and square-roots that one
needs to memorize. These numbers are special because
many mathematics exercises utilize square numbers.

23. Equations of the form x2 = c has two answers:
x = +c or –c if c>0.
Radicals and Pythagorean Theorem

24. Equations of the form x2 = c has two answers:
x = +c or –c if c>0.
Radicals and Pythagorean Theorem
Example B. Solve the following equations.
a. x2 = 25

25. Equations of the form x2 = c has two answers:
x = +c or –c if c>0.
Radicals and Pythagorean Theorem
Example B. Solve the following equations.
a. x2 = 25
x = ±25

26. Equations of the form x2 = c has two answers:
x = +c or –c if c>0.
Radicals and Pythagorean Theorem
Example B. Solve the following equations.
a. x2 = 25
x = ±25 = ±5

27. Equations of the form x2 = c has two answers:
x = +c or –c if c>0.
Radicals and Pythagorean Theorem
Example B. Solve the following equations.
a. x2 = 25
x = ±25 = ±5
b. x2 = –4

28. Equations of the form x2 = c has two answers:
x = +c or –c if c>0.
Radicals and Pythagorean Theorem
Example B. Solve the following equations.
a. x2 = 25
x = ±25 = ±5
b. x2 = –4
Solution does not exist.

29. Equations of the form x2 = c has two answers:
x = +c or –c if c>0.
Radicals and Pythagorean Theorem
Example B. Solve the following equations.
a. x2 = 25
x = ±25 = ±5
b. x2 = –4
Solution does not exist.
c. x2 = 8

30. Equations of the form x2 = c has two answers:
x = +c or –c if c>0.
Radicals and Pythagorean Theorem
Example B. Solve the following equations.
a. x2 = 25
x = ±25 = ±5
b. x2 = –4
Solution does not exist.
c. x2 = 8
x = ±8

31. Equations of the form x2 = c has two answers:
x = +c or –c if c>0.
Radicals and Pythagorean Theorem
Example B. Solve the following equations.
a. x2 = 25
x = ±25 = ±5
b. x2 = –4
Solution does not exist.
c. x2 = 8
x = ±8  ±2.8284.. by calculator

32. Equations of the form x2 = c has two answers:
x = +c or –c if c>0.
Radicals and Pythagorean Theorem
Example B. Solve the following equations.
a. x2 = 25
x = ±25 = ±5
b. x2 = –4
Solution does not exist.
c. x2 = 8
x = ±8  ±2.8284.. by calculator
exact answer approximate answer

33. Equations of the form x2 = c has two answers:
x = +c or –c if c>0.
Radicals and Pythagorean Theorem
Example B. Solve the following equations.
a. x2 = 25
x = ±25 = ±5
b. x2 = –4
Solution does not exist.
c. x2 = 8
x = ±8  ±2.8284.. by calculator
exact answer approximate answer
In geometry square roots show up in distance problems
because of Pythagorean theorem.

34. A right triangle is a triangle with a right angle as one of its
angle.
Radicals and Pythagorean Theorem

35. A right triangle is a triangle with a right angle as one of its
angles. The longest side C of a right triangle is called the
hypotenuse,
Radicals and Pythagorean Theorem
hypotenuse
C

36. A right triangle is a triangle with a right angle as one of its
angles. The longest side C of a right triangle is called the
hypotenuse, the two sides A and B forming the right angle
are called the legs.
Radicals and Pythagorean Theorem
hypotenuse
legs
A
B
C

37. A right triangle is a triangle with a right angle as one of its
angles. The longest side C of a right triangle is called the
hypotenuse, the two sides A and B forming the right angle
are called the legs.
Pythagorean Theorem
Given a right triangle with labeling as shown, then
A2 + B2 = C2
Radicals and Pythagorean Theorem
hypotenuse
legs
A
B
C

38. Radicals and Pythagorean Theorem
There are two types of problems that use the Pythagorean
Theorem to find the sides of the right triangles

39. Radicals and Pythagorean Theorem
There are two types of problems that use the Pythagorean
Theorem to find the sides of the right triangles-finding the
hypotenuse versus finding the legs.

40. Example C. Find the missing side of the following right
triangles. Find the exact answer and the approximate answer.
Draw.
a. a = 5, b = 12, c = ?
Radicals and Pythagorean Theorem
There are two types of problems that use the Pythagorean
Theorem to find the sides of the right triangles-finding the
hypotenuse versus finding the legs.

41. Example C. Find the missing side of the following right
triangles. Find the exact answer and the approximate answer.
Draw.
a. a = 5, b = 12, c = ?
Radicals and Pythagorean Theorem
There are two types of problems that use the Pythagorean
Theorem to find the sides of the right triangles-finding the
hypotenuse versus finding the legs.

42. Example C. Find the missing side of the following right
triangles. Find the exact answer and the approximate answer.
Draw.
a. a = 5, b = 12, c = ?
Since it is a right triangle,
122 + 52 = c2
Radicals and Pythagorean Theorem
There are two types of problems that use the Pythagorean
Theorem to find the sides of the right triangles-finding the
hypotenuse versus finding the legs.

43. Example C. Find the missing side of the following right
triangles. Find the exact answer and the approximate answer.
Draw.
a. a = 5, b = 12, c = ?
Since it is a right triangle,
122 + 52 = c2
144 + 25 = c2
Radicals and Pythagorean Theorem
There are two types of problems that use the Pythagorean
Theorem to find the sides of the right triangles-finding the
hypotenuse versus finding the legs.

44. Example C. Find the missing side of the following right
triangles. Find the exact answer and the approximate answer.
Draw.
a. a = 5, b = 12, c = ?
Since it is a right triangle,
122 + 52 = c2
144 + 25 = c2
169 = c2
Radicals and Pythagorean Theorem
There are two types of problems that use the Pythagorean
Theorem to find the sides of the right triangles-finding the
hypotenuse versus finding the legs.

45. Example C. Find the missing side of the following right
triangles. Find the exact answer and the approximate answer.
Draw.
a. a = 5, b = 12, c = ?
Since it is a right triangle,
122 + 52 = c2
144 + 25 = c2
169 = c2
So c = ±169 = ±13
Radicals and Pythagorean Theorem
There are two types of problems that use the Pythagorean
Theorem to find the sides of the right triangles-finding the
hypotenuse versus finding the legs.

46. Example C. Find the missing side of the following right
triangles. Find the exact answer and the approximate answer.
Draw.
a. a = 5, b = 12, c = ?
Since it is a right triangle,
122 + 52 = c2
144 + 25 = c2
169 = c2
So c = ±169 = ±13
Since length can’t be
negative, therefore c = 13.
Radicals and Pythagorean Theorem
There are two types of problems that use the Pythagorean
Theorem to find the sides of the right triangles-finding the
hypotenuse versus finding the legs.

47. b. a = 5, c = 12, b = ?
Radicals and Pythagorean Theorem

48. b. a = 5, c = 12, b = ?
Radicals and Pythagorean Theorem

49. b. a = 5, c = 12, b = ?
Since it is a right triangle,
b2 + 52 = 122
Radicals and Pythagorean Theorem

50. b. a = 5, c = 12, b = ?
Since it is a right triangle,
b2 + 52 = 122
b2 + 25 = 144
Radicals and Pythagorean Theorem

51. b. a = 5, c = 12, b = ?
Since it is a right triangle,
b2 + 52 = 122
b2 + 25 = 144
b2 = 144 – 25
Radicals and Pythagorean Theorem

52. b. a = 5, c = 12, b = ?
Since it is a right triangle,
b2 + 52 = 122
b2 + 25 = 144
b2 = 144 – 25
So b = ±119  ±10.9.
Radicals and Pythagorean Theorem

53. b. a = 5, c = 12, b = ?
Since it is a right triangle,
b2 + 52 = 122
b2 + 25 = 144
b2 = 144 – 25
So b = ±119  ±10.9.
But length can’t be negative,
therefore b = 119  10.9
Radicals and Pythagorean Theorem

54. b. a = 5, c = 12, b = ?
Since it is a right triangle,
b2 + 52 = 122
b2 + 25 = 144
b2 = 144 – 25
So b = ±119  ±10.9.
But length can’t be negative,
therefore b = 119  10.9
Radicals and Pythagorean Theorem
The Distance Formula

55. b. a = 5, c = 12, b = ?
Since it is a right triangle,
b2 + 52 = 122
b2 + 25 = 144
b2 = 144 – 25
So b = ±119  ±10.9.
But length can’t be negative,
therefore b = 119  10.9
Radicals and Pythagorean Theorem
The Distance Formula

56. b. a = 5, c = 12, b = ?
Since it is a right triangle,
b2 + 52 = 122
b2 + 25 = 144
b2 = 144 – 25
So b = ±119  ±10.9.
But length can’t be negative,
therefore b = 119  10.9
Radicals and Pythagorean Theorem
Let (x1, y1) and (x2, y2) be two
points, D = distance between them,
The Distance Formula
(x1, y1)
(x2, y2)
D

57. b. a = 5, c = 12, b = ?
Since it is a right triangle,
b2 + 52 = 122
b2 + 25 = 144
b2 = 144 – 25
So b = ±119  ±10.9.
But length can’t be negative,
therefore b = 119  10.9
Radicals and Pythagorean Theorem
Let (x1, y1) and (x2, y2) be two
points, D = distance between them,
then D2 = Δx2 + Δy2
The Distance Formula
(x1, y1)
(x2, y2)
Δy
Δx
D

58. b. a = 5, c = 12, b = ?
Since it is a right triangle,
b2 + 52 = 122
b2 + 25 = 144
b2 = 144 – 25
So b = ±119  ±10.9.
But length can’t be negative,
therefore b = 119  10.9
Radicals and Pythagorean Theorem
Let (x1, y1) and (x2, y2) be two
points, D = distance between them,
then D2 = Δx2 + Δy2 where
The Distance Formula
(x1, y1)
(x2, y2)
Δx = x2 – x1
Δy
Δx = x2 – x1
D

59. b. a = 5, c = 12, b = ?
Since it is a right triangle,
b2 + 52 = 122
b2 + 25 = 144
b2 = 144 – 25
So b = ±119  ±10.9.
But length can’t be negative,
therefore b = 119  10.9
Radicals and Pythagorean Theorem
Let (x1, y1) and (x2, y2) be two
points, D = distance between them,
then D2 = Δx2 + Δy2 where
The Distance Formula
(x1, y1)
(x2, y2)
Δx = x2 – x1 Δy = y2 – y1and
Δy = y2 – y1
Δx = x2 – x1
D
by the Pythagorean Theorem.

60. b. a = 5, c = 12, b = ?
Since it is a right triangle,
b2 + 52 = 122
b2 + 25 = 144
b2 = 144 – 25
So b = ±119  ±10.9.
But length can’t be negative,
therefore b = 119  10.9
Radicals and Pythagorean Theorem
Let (x1, y1) and (x2, y2) be two
points, D = distance between them,
then D2 = Δx2 + Δy2 where
The Distance Formula
(x1, y1)
(x2, y2)
Δx = x2 – x1 Δy = y2 – y1and
Δy = y2 – y1
Δx = x2 – x1by the Pythagorean Theorem.
Hence we’ve the Distant Formula:
D = √ Δx2 + Δy2
D

61. b. a = 5, c = 12, b = ?
Since it is a right triangle,
b2 + 52 = 122
b2 + 25 = 144
b2 = 144 – 25
So b = ±119  ±10.9.
But length can’t be negative,
therefore b = 119  10.9
Radicals and Pythagorean Theorem
Let (x1, y1) and (x2, y2) be two
points, D = distance between them,
then D2 = Δx2 + Δy2 where
The Distance Formula
(x1, y1)
(x2, y2)
Δx = x2 – x1 Δy = y2 – y1and
Δy = y2 – y1
Δx = x2 – x1by the Pythagorean Theorem.
Hence we’ve the Distant Formula:
D = √ Δx2 + Δy2 = √ (x2 – x1)2 + (y2 – y1)2
D

62. Example D. Find the distance between (–1, 3) and (2, –4).
(-1, 3)
– ( 2, -4)
D
Radicals and Pythagorean Theorem
–3, 7
Δx Δy

63. Example D. Find the distance between (–1, 3) and (2, –4).
(-1, 3)
– ( 2, -4)
D
Radicals and Pythagorean Theorem
–3, 7
Δx Δy

64. Example D. Find the distance between (–1, 3) and (2, –4).
(-1, 3)
– ( 2, -4)
D
Radicals and Pythagorean Theorem
–3, 7
Δx Δy

65. Example D. Find the distance between (–1, 3) and (2, –4).
(-1, 3)
– ( 2, -4)
D
Radicals and Pythagorean Theorem
–3, 7
Δx Δy

66. Example D. Find the distance between (–1, 3) and (2, –4).
(-1, 3)
– ( 2, -4)
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy

67. Example D. Find the distance between (–1, 3) and (2, –4).
(-1, 3)
– ( 2, -4)
D = (–3)2 + 72 = 58  7.62
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy

68. Example D. Find the distance between (–1, 3) and (2, –4).
(-1, 3)
– ( 2, -4)
D = (–3)2 + 72 = 58  7.62
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
Higher Root

69. Example D. Find the distance between (–1, 3) and (2, –4).
(-1, 3)
– ( 2, -4)
D = (–3)2 + 72 = 58  7.62
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
Higher Root
If r3 = x, then we say a is the cube root of x.

70. Example D. Find the distance between (–1, 3) and (2, –4).
(-1, 3)
– ( 2, -4)
D = (–3)2 + 72 = 58  7.62
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
Higher Root
If r3 = x, then we say a is the cube root of x. We write this as
x = r.3

71. Example D. Find the distance between (–1, 3) and (2, –4).
(-1, 3)
– ( 2, -4)
D = (–3)2 + 72 = 58  7.62
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
Higher Root
If r3 = x, then we say a is the cube root of x. We write this as
x = r. In general, if r k = x, then we say r is the k’th root of x,3

72. Example D. Find the distance between (–1, 3) and (2, –4).
(-1, 3)
– ( 2, -4)
D = (–3)2 + 72 = 58  7.62
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
Higher Root
If r3 = x, then we say a is the cube root of x. We write this as
x = r. In general, if r k = x, then we say r is the k’th root of x,
and we write it as a = x.
3
k

73. Example D. Find the distance between (–1, 3) and (2, –4).
(-1, 3)
– ( 2, -4)
D = (–3)2 + 72 = 58  7.62
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
Higher Root
If r3 = x, then we say a is the cube root of x. We write this as
x = r. In general, if r k = x, then we say r is the k’th root of x,
and we write it as a = x. In the cases of even roots,
i.e. k = 2, 4, 6, … we must have x > 0 and that x = a > 0.
3
k
k

74. Example D. Find the distance between (–1, 3) and (2, –4).
(-1, 3)
– ( 2, -4)
D = (–3)2 + 72 = 58  7.62
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
3
k
k
Example E.
a. 8 =
3
Higher Root
If r3 = x, then we say a is the cube root of x. We write this as
x = r. In general, if r k = x, then we say r is the k’th root of x,
and we write it as a = x. In the cases of even roots,
i.e. k = 2, 4, 6, … we must have x > 0 and that x = a > 0.

75. Example D. Find the distance between (–1, 3) and (2, –4).
(-1, 3)
– ( 2, -4)
D = (–3)2 + 72 = 58  7.62
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
3
k
k
Example E.
a. 8 = 2
3
Higher Root
If r3 = x, then we say a is the cube root of x. We write this as
x = r. In general, if r k = x, then we say r is the k’th root of x,
and we write it as a = x. In the cases of even roots,
i.e. k = 2, 4, 6, … we must have x > 0 and that x = a > 0.

76. Example D. Find the distance between (–1, 3) and (2, –4).
(-1, 3)
– ( 2, -4)
D = (–3)2 + 72 = 58  7.62
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
3
k
k
Example E.
a. 8 = 2 b. –1 =
3 3
Higher Root
If r3 = x, then we say a is the cube root of x. We write this as
x = r. In general, if r k = x, then we say r is the k’th root of x,
and we write it as a = x. In the cases of even roots,
i.e. k = 2, 4, 6, … we must have x > 0 and that x = a > 0.

77. Example D. Find the distance between (–1, 3) and (2, –4).
(-1, 3)
– ( 2, -4)
D = (–3)2 + 72 = 58  7.62
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
3
k
k
Example E.
a. 8 = 2 b. –1 = –1
3 3
Higher Root
If r3 = x, then we say a is the cube root of x. We write this as
x = r. In general, if r k = x, then we say r is the k’th root of x,
and we write it as a = x. In the cases of even roots,
i.e. k = 2, 4, 6, … we must have x > 0 and that x = a > 0.

78. Example D. Find the distance between (–1, 3) and (2, –4).
(-1, 3)
– ( 2, -4)
D = (–3)2 + 72 = 58  7.62
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
3
k
k
Example E.
a. 8 = 2 b. –1 = –1 c.–27 =
3 3 3
Higher Root
If r3 = x, then we say a is the cube root of x. We write this as
x = r. In general, if r k = x, then we say r is the k’th root of x,
and we write it as a = x. In the cases of even roots,
i.e. k = 2, 4, 6, … we must have x > 0 and that x = a > 0.

79. Example D. Find the distance between (–1, 3) and (2, –4).
(-1, 3)
– ( 2, -4)
D = (–3)2 + 72 = 58  7.62
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
3
k
k
Example E.
a. 8 = 2 b. –1 = –1 c.–27 = –3
3 3 3
Higher Root
If r3 = x, then we say a is the cube root of x. We write this as
x = r. In general, if r k = x, then we say r is the k’th root of x,
and we write it as a = x. In the cases of even roots,
i.e. k = 2, 4, 6, … we must have x > 0 and that x = a > 0.

80. Example D. Find the distance between (–1, 3) and (2, –4).
(-1, 3)
– ( 2, -4)
D = (–3)2 + 72 = 58  7.62
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
3
k
k
Example E.
a. 8 = 2 b. –1 = –1 c.–27 = –3
d. 16 =
3 3 3
Higher Root
If r3 = x, then we say a is the cube root of x. We write this as
x = r. In general, if r k = x, then we say r is the k’th root of x,
and we write it as a = x. In the cases of even roots,
i.e. k = 2, 4, 6, … we must have x > 0 and that x = a > 0.
4

81. Example D. Find the distance between (–1, 3) and (2, –4).
(-1, 3)
– ( 2, -4)
D = (–3)2 + 72 = 58  7.62
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
3
k
k
Example E.
a. 8 = 2 b. –1 = –1 c.–27 = –3
d. 16 = 2
3 3 3
Higher Root
If r3 = x, then we say a is the cube root of x. We write this as
x = r. In general, if r k = x, then we say r is the k’th root of x,
and we write it as a = x. In the cases of even roots,
i.e. k = 2, 4, 6, … we must have x > 0 and that x = a > 0.
4

82. Example D. Find the distance between (–1, 3) and (2, –4).
(-1, 3)
– ( 2, -4)
D = (–3)2 + 72 = 58  7.62
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
3
k
k
Example E.
a. 8 = 2 b. –1 = –1 c.–27 = –3
d. 16 = 2 e. –16 =
3 3 3
Higher Root
If r3 = x, then we say a is the cube root of x. We write this as
x = r. In general, if r k = x, then we say r is the k’th root of x,
and we write it as a = x. In the cases of even roots,
i.e. k = 2, 4, 6, … we must have x > 0 and that x = a > 0.
4 4

83. Example D. Find the distance between (–1, 3) and (2, –4).
(-1, 3)
– ( 2, -4)
D = (–3)2 + 72 = 58  7.62
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
3
k
k
Example E.
a. 8 = 2 b. –1 = –1 c.–27 = –3
d. 16 = 2 e. –16 = not real
3 3 3
Higher Root
If r3 = x, then we say a is the cube root of x. We write this as
x = r. In general, if r k = x, then we say r is the k’th root of x,
and we write it as a = x. In the cases of even roots,
i.e. k = 2, 4, 6, … we must have x > 0 and that x = a > 0.
4 4

84. Example D. Find the distance between (–1, 3) and (2, –4).
(-1, 3)
– ( 2, -4)
D = (–3)2 + 72 = 58  7.62
D
7
–3
Radicals and Pythagorean Theorem
–3, 7
Δx Δy
3
k
k
Example E.
a. 8 = 2 b. –1 = –1 c.–27 = –3
d. 16 = 2 e. –16 = not real f. 10 ≈ 2.15..
3 3 3
Higher Root
If r3 = x, then we say a is the cube root of x. We write this as
x = r. In general, if r k = x, then we say r is the k’th root of x,
and we write it as a = x. In the cases of even roots,
i.e. k = 2, 4, 6, … we must have x > 0 and that x = a > 0.
4 4 3

85. Radicals and Pythagorean Theorem
Rational and Irrational Numbers

86. The number 2 is the length of the
hypotenuse of the right triangle as shown.
Radicals and Pythagorean Theorem
Rational and Irrational Numbers
2
1
1

87. The number 2 is the length of the
hypotenuse of the right triangle as shown.
Radicals and Pythagorean Theorem
Rational and Irrational Numbers
2
1
1
It can be shown that 2 can not be
represented as a ratio of whole numbers i.e.
P/Q, where P and Q are integers.

88. The number 2 is the length of the
hypotenuse of the right triangle as shown.
Radicals and Pythagorean Theorem
Rational and Irrational Numbers
2
1
1
It can be shown that 2 can not be
represented as a ratio of whole numbers i.e.
P/Q, where P and Q are integers.
Hence these numbers are called irrational (non–ratio)
numbers.

89. The number 2 is the length of the
hypotenuse of the right triangle as shown.
Radicals and Pythagorean Theorem
Rational and Irrational Numbers
2
1
1
It can be shown that 2 can not be
represented as a ratio of whole numbers i.e.
P/Q, where P and Q are integers.
Hence these numbers are called irrational (non–ratio)
numbers. Most real numbers are irrational, not fractions, i.e.
they can’t be represented as ratios of two integers.

90. The number 2 is the length of the
hypotenuse of the right triangle as shown.
Radicals and Pythagorean Theorem
Rational and Irrational Numbers
2
1
1
It can be shown that 2 can not be
represented as a ratio of whole numbers i.e.
P/Q, where P and Q are integers.
Hence these numbers are called irrational (non–ratio)
numbers. Most real numbers are irrational, not fractions, i.e.
they can’t be represented as ratios of two integers. The real
line is populated sparsely by fractional locations.

91. The number 2 is the length of the
hypotenuse of the right triangle as shown.
Radicals and Pythagorean Theorem
Rational and Irrational Numbers
2
1
1
It can be shown that 2 can not be
represented as a ratio of whole numbers i.e.
P/Q, where P and Q are integers.
Hence these numbers are called irrational (non–ratio)
numbers. Most real numbers are irrational, not fractions, i.e.
they can’t be represented as ratios of two integers. The real
line is populated sparsely by fractional locations. The
Pythagorean school of the ancient Greeks had believed that
all the measurable quantities in the universe are fractional
quantities. The “discovery” of these extra irrational numbers
caused a profound intellectual crisis.

92. The number 2 is the length of the
hypotenuse of the right triangle as shown.
Radicals and Pythagorean Theorem
Rational and Irrational Numbers
2
1
1
It can be shown that 2 can not be
represented as a ratio of whole numbers i.e.
P/Q, where P and Q are integers.
Hence these numbers are called irrational (non–ratio)
numbers. Most real numbers are irrational, not fractions, i.e.
they can’t be represented as ratios of two integers. The real
line is populated sparsely by fractional locations. The
Pythagorean school of the ancient Greeks had believed that
all the measurable quantities in the universe are fractional
quantities. The “discovery” of these extra irrational numbers
caused a profound intellectual crisis. It wasn’t until the last two
centuries that mathematicians clarified the strange questions
“How many and what kind of numbers are there?”

93. Radicals and Pythagorean Theorem
Exercise A. Solve for x. Give both the exact and approximate
answers. If the answer does not exist, state so.
1. x2 = 1 2. x2 – 5 = 4 3. x2 + 5 = 4
4. 2x2 = 31 5. 4x2 – 5 = 4 6. 5 = 3x2 + 1
7. 4x2 = 1 8. x2 – 32 = 42 9. x2 + 62 = 102
10. 2x2 + 7 = 11 11. 2x2 – 5 = 6 12. 4 = 3x2 + 5
x
3
4
Exercise B. Solve for x. Give both the exact and approximate
answers. If the answer does not exist, state so.
13. 4
3
x14. x
12
515.
x
1
116. 2
1
x17. 3 2
3
x18.

94. Radicals and Pythagorean Theorem
x
4
19.
x
x20.
3 /3
21.
43 5 2
6 /3
Exercise C. Given the following information, find the rise and
run from A to B i.e. Δx and Δy. Find the distance from A to B.
A
22.
B
A
23.
B
24. A = (2, –3) , B = (1, 5) 25. A = (1, 5) , B = (2, –3)
26. A = (–2 , –5) , B = (3, –2) 27. A = (–4 , –1) , B = (2, –3)
28. Why is the distance from A to B the same as from B to A?

95. Exercise D. Find the exact answer.
Radicals and Pythagorean Theorem
3
–129. 30. 13
–12534.
8 31. –13
8
3
–2732.
33. –13
64
3
10035. 100036.
3
10,00037. 1,000,00038.
3
0.0139. 0.00140.
3
0.000141. 0.00000142.
3