pythagorean theorem and square roots

1. Pythagorean Theorem and Square Roots

2. A right triangle is a triangle with a right angle as one of its
angle.
Pythagorean Theorem and Square Roots

3. 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,
Pythagorean Theorem and Square Roots
hypotenuse
C

4. 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 and Square Roots
hypotenuse
legs
A
B
C

5. 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 as shown and A, B, and C
be the length of the sides, then A2 + B2 = C2.
Pythagorean Theorem and Square Roots
hypotenuse
legs
A
B
C

6. Pythagorean Theorem
Given a right triangle
with labeling as shown,
then A2 + B2 = C2
Pythagorean Theorem and Square Roots
Pythagorean Theorem allows us
to compute a length, i.e. a distance,
without measuring it directly.

7. Pythagorean Theorem
Given a right triangle
with labeling as shown,
then A2 + B2 = C2
Pythagorean Theorem and Square Roots
Pythagorean Theorem allows us
to compute a length, i.e. a distance,
without measuring it directly.
Example A. A 5–meter ladder leans
against a wall as shown. Its base is
3 meters from the wall. How high is
the wall?
5 m
3 m
?

8. Pythagorean Theorem
Given a right triangle
with labeling as shown,
then A2 + B2 = C2
Pythagorean Theorem and Square Roots
Pythagorean Theorem allows us
to compute a length, i.e. a distance,
without measuring it directly.
Example A. A 5–meter ladder leans
against a wall as shown. Its base is
3 meters from the wall. How high is
the wall?
5 m
3 m
? = h
Let h be the height of the wall.

9. Pythagorean Theorem
Given a right triangle
with labeling as shown,
then A2 + B2 = C2
Pythagorean Theorem and Square Roots
Pythagorean Theorem allows us
to compute a length, i.e. a distance,
without measuring it directly.
Example A. A 5–meter ladder leans
against a wall as shown. Its base is
3 meters from the wall. How high is
the wall?
5 m
3 m
? = h
Let h be the height of the wall.
The wall and the ground form a right triangle,
hence by the Pythagorean Theorem
we have that h2 + 32 = 52

10. Pythagorean Theorem
Given a right triangle
with labeling as shown,
then A2 + B2 = C2
Pythagorean Theorem and Square Roots
Pythagorean Theorem allows us
to compute a length, i.e. a distance,
without measuring it directly.
Example A. A 5–meter ladder leans
against a wall as shown. Its base is
3 meters from the wall. How high is
the wall?
5 m
3 m
? = h
Let h be the height of the wall.
The wall and the ground form a right triangle,
hence by the Pythagorean Theorem
we have that h2 + 32 = 52
h2 + 9 = 25

11. Pythagorean Theorem
Given a right triangle
with labeling as shown,
then A2 + B2 = C2
Pythagorean Theorem and Square Roots
Pythagorean Theorem allows us
to compute a length, i.e. a distance,
without measuring it directly.
Example A. A 5–meter ladder leans
against a wall as shown. Its base is
3 meters from the wall. How high is
the wall?
5 m
3 m
? = h
Let h be the height of the wall.
The wall and the ground form a right triangle,
hence by the Pythagorean Theorem
we have that h2 + 32 = 52
h2 + 9 = 25
–9 –9
subtract 9
from both sides

12. Pythagorean Theorem
Given a right triangle
with labeling as shown,
then A2 + B2 = C2
Pythagorean Theorem and Square Roots
Pythagorean Theorem allows us
to compute a length, i.e. a distance,
without measuring it directly.
Example A. A 5–meter ladder leans
against a wall as shown. Its base is
3 meters from the wall. How high is
the wall?
5 m
3 m
? = h
Let h be the height of the wall.
The wall and the ground form a right triangle,
hence by the Pythagorean Theorem
we have that h2 + 32 = 52
h2 + 9 = 25
–9 –9
h2 = 16
subtract 9
from both sides

13. Pythagorean Theorem
Given a right triangle
with labeling as shown,
then A2 + B2 = C2
Pythagorean Theorem and Square Roots
Pythagorean Theorem allows us
to compute a length, i.e. a distance,
without measuring it directly.
Example A. A 5–meter ladder leans
against a wall as shown. Its base is
3 meters from the wall. How high is
the wall?
5 m
3 m
? = h
Let h be the height of the wall.
The wall and the ground form a right triangle,
hence by the Pythagorean Theorem
we have that h2 + 32 = 52
h2 + 9 = 25
–9 –9
h2 = 16
By trying different numbers for h, we find that 42 = 16
so h = 4 or that the wall is 4–meter high.
subtract 9
from both sides

14. Pythagorean Theorem and Square Roots
Square Root

15. Pythagorean Theorem and Square Roots
Square Root
From example A, we encountered that “the square of 4 is16”:
4 16
(#)2

16. We also state this relation as “the square–root of 16 is 4”,
i.e. 4 is the source for output “16”,
Pythagorean Theorem and Square Roots
Square Root
From example A, we encountered that “the square of 4 is16”:
4 16
(#)2

17. We also state this relation as “the square–root of 16 is 4”,
i.e. 4 is the source for output “16”, and it’s written as 16 = 4:
Pythagorean Theorem and Square Roots
Square Root
From example A, we encountered that “the square of 4 is16”:
4 16
(#)2

18. We also state this relation as “the square–root of 16 is 4”,
i.e. 4 is the source for output “16”, and it’s written as 16 = 4:
Pythagorean Theorem and Square Roots
Square Root
From example A, we encountered that “the square of 4 is16”:
4 16
(#)2
16 = 4 16
 #

19. We also state this relation as “the square–root of 16 is 4”,
i.e. 4 is the source for output “16”, and it’s written as 16 = 4:
Pythagorean Theorem and Square Roots
Note that both +4 and –4, when squared, give 16.
Square Root
From example A, we encountered that “the square of 4 is16”:
4 16
(#)2
16 = 4 16
 #

20. We also state this relation as “the square–root of 16 is 4”,
i.e. 4 is the source for output “16”, and it’s written as 16 = 4:
Pythagorean Theorem and Square Roots
Note that both +4 and –4, when squared, give 16. But we
designate the “square root of 16” i.e. 16 or sqrt(16) to be +4.
Square Root
From example A, we encountered that “the square of 4 is16”:
4 16
(#)2
16 = 4 16
 #

21. We also state this relation as “the square–root of 16 is 4”,
i.e. 4 is the source for output “16”, and it’s written as 16 = 4:
Pythagorean Theorem and Square Roots
Note that both +4 and –4, when squared, give 16. But we
designate the “square root of 16” i.e. 16 or sqrt(16) to be +4.
We refer “–4” as the “negative of the square root of 16”.
Square Root
From example A, we encountered that “the square of 4 is16”:
4 16
(#)2
16 = 4 16
 #

22. We also state this relation as “the square–root of 16 is 4”,
i.e. 4 is the source for output “16”, and it’s written as 16 = 4:
Pythagorean Theorem and Square Roots
Definition: If a2 → x and a is not negative, then a is called the
square root of x.
Note that both +4 and –4, when squared, give 16. But we
designate the “square root of 16” i.e. 16 or sqrt(16) to be +4.
We refer “–4” as the “negative of the square root of 16”.
Square Root
From example A, we encountered that “the square of 4 is16”:
4 16
(#)2
16 = 4 16
 #

23. We also state this relation as “the square–root of 16 is 4”,
i.e. 4 is the source for output “16”, and it’s written as 16 = 4:
Pythagorean Theorem and Square Roots
Definition: If a2 → x and a is not negative, then a is called the
square root of x. This is written as sqrt(x) = a, or x = a.
Note that both +4 and –4, when squared, give 16. But we
designate the “square root of 16” i.e. 16 or sqrt(16) to be +4.
We refer “–4” as the “negative of the square root of 16”.
Square Root
From example A, we encountered that “the square of 4 is16”:
4 16
(#)2
16 = 4 16
 #

24. We also state this relation as “the square–root of 16 is 4”,
i.e. 4 is the source for output “16”, and it’s written as 16 = 4:
Example A.
a. Sqrt(16) =
c.3 =
Pythagorean Theorem and Square Roots
Definition: If a2 → x and a is not negative, then a is called the
square root of x. This is written as sqrt(x) = a, or x = a.
b. 1/9 =
d. –3 =
Note that both +4 and –4, when squared, give 16. But we
designate the “square root of 16” i.e. 16 or sqrt(16) to be +4.
We refer “–4” as the “negative of the square root of 16”.
Square Root
From example A, we encountered that “the square of 4 is16”:
4 16
(#)2
16 = 4 16
 #

25. We also state this relation as “the square–root of 16 is 4”,
i.e. 4 is the source for output “16”, and it’s written as 16 = 4:
Example A.
a. Sqrt(16) = 4
c.3 =
Pythagorean Theorem and Square Roots
Definition: If a2 → x and a is not negative, then a is called the
square root of x. This is written as sqrt(x) = a, or x = a.
b. 1/9 =
d. –3 =
Note that both +4 and –4, when squared, give 16. But we
designate the “square root of 16” i.e. 16 or sqrt(16) to be +4.
We refer “–4” as the “negative of the square root of 16”.
Square Root
From example A, we encountered that “the square of 4 is16”:
4 16
(#)2
16 = 4 16
 #

26. We also state this relation as “the square–root of 16 is 4”,
i.e. 4 is the source for output “16”, and it’s written as 16 = 4:
Example A.
a. Sqrt(16) = 4
c.3 =
Pythagorean Theorem and Square Roots
Definition: If a2 → x and a is not negative, then a is called the
square root of x. This is written as sqrt(x) = a, or x = a.
b. 1/9 = 1/3
d. –3 =
Note that both +4 and –4, when squared, give 16. But we
designate the “square root of 16” i.e. 16 or sqrt(16) to be +4.
We refer “–4” as the “negative of the square root of 16”.
Square Root
From example A, we encountered that “the square of 4 is16”:
4 16
(#)2
16 = 4 16
 #

27. We also state this relation as “the square–root of 16 is 4”,
i.e. 4 is the source for output “16”, and it’s written as 16 = 4:
Example A.
a. Sqrt(16) = 4
c.3 = 1.732.. by calculator
or that 3 ≈ 1.7 (approx.)
Pythagorean Theorem and Square Roots
Definition: If a2 → x and a is not negative, then a is called the
square root of x. This is written as sqrt(x) = a, or x = a.
b. 1/9 = 1/3
d. –3 =
Note that both +4 and –4, when squared, give 16. But we
designate the “square root of 16” i.e. 16 or sqrt(16) to be +4.
We refer “–4” as the “negative of the square root of 16”.
Square Root
From example A, we encountered that “the square of 4 is16”:
4 16
(#)2
16 = 4 16
 #

28. We also state this relation as “the square–root of 16 is 4”,
i.e. 4 is the source for output “16”, and it’s written as 16 = 4:
Example A.
a. Sqrt(16) = 4
c.3 = 1.732.. by calculator
or that 3 ≈ 1.7 (approx.)
Pythagorean Theorem and Square Roots
Definition: If a2 → x and a is not negative, then a is called the
square root of x. This is written as sqrt(x) = a, or x = a.
b. 1/9 = 1/3
d. –3 = doesn’t exist (why?),
and the calculator returns “Error”.
Note that both +4 and –4, when squared, give 16. But we
designate the “square root of 16” i.e. 16 or sqrt(16) to be +4.
We refer “–4” as the “negative of the square root of 16”.
Square Root
From example A, we encountered that “the square of 4 is16”:
4 16
(#)2
16 = 4 16
 #

29. 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
Pythagorean Theorem and Square Roots
Following are the square numbers and square-roots that one
needs to memorize.

30. 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
Pythagorean Theorem and Square Roots
Following are the square numbers and square-roots that one
needs to memorize. These numbers are special because
many mathematics exercises utilize square numbers.

31. 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.
Pythagorean Theorem and Square Roots
Following are the square numbers and square-roots that one
needs to memorize. These numbers are special because
many mathematics exercises utilize square numbers.

32. 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
Pythagorean Theorem and Square Roots
Following are the square numbers and square-roots that one
needs to memorize. These numbers are special because
many mathematics exercises utilize square numbers.

33. 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
Pythagorean Theorem and Square Roots
Following are the square numbers and square-roots that one
needs to memorize. These numbers are special because
many mathematics exercises utilize square numbers.

34. 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
Pythagorean Theorem and Square Roots
Following are the square numbers and square-roots that one
needs to memorize. These numbers are special because
many mathematics exercises utilize square numbers.

35. 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,
Pythagorean Theorem and Square Roots
Following are the square numbers and square-roots that one
needs to memorize. These numbers are special because
many mathematics exercises utilize square numbers.

36. 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.
Pythagorean Theorem and Square Roots
Following are the square numbers and square-roots that one
needs to memorize. These numbers are special because
many mathematics exercises utilize square numbers.

37. 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….
Pythagorean Theorem and Square Roots
Following are the square numbers and square-roots that one
needs to memorize. These numbers are special because
many mathematics exercises utilize square numbers.

38. Pythagorean Theorem and Square Roots
Depending on which is the missing side, there are two versions
of calculation based on the Pythagorean Theorem –
finding the hypotenuse versus finding a leg.

39. Pythagorean Theorem and Square Roots
Depending on which is the missing side, there are two versions
of calculation based on the Pythagorean Theorem –
finding the hypotenuse versus finding a leg.
Example B.
Find the missing side of the following right triangles.

40. a. We have the legs a = 5, b = 12,
Pythagorean Theorem and Square Roots
Depending on which is the missing side, there are two versions
of calculation based on the Pythagorean Theorem –
finding the hypotenuse versus finding a leg.
Example B.
Find the missing side of the following right triangles.
we are to find the hypotenuse,

41. a. We have the legs a = 5, b = 12,
Pythagorean Theorem and Square Roots
Depending on which is the missing side, there are two versions
of calculation based on the Pythagorean Theorem –
finding the hypotenuse versus finding a leg.
Example B.
Find the missing side of the following right triangles.
we are to find the hypotenuse,
so 122 + 52 = c2

42. a. We have the legs a = 5, b = 12,
Pythagorean Theorem and Square Roots
Depending on which is the missing side, there are two versions
of calculation based on the Pythagorean Theorem –
finding the hypotenuse versus finding a leg.
Example B.
Find the missing side of the following right triangles.
we are to find the hypotenuse,
so 122 + 52 = c2
144 + 25 = c2

43. a. We have the legs a = 5, b = 12,
Pythagorean Theorem and Square Roots
Depending on which is the missing side, there are two versions
of calculation based on the Pythagorean Theorem –
finding the hypotenuse versus finding a leg.
Example B.
Find the missing side of the following right triangles.
we are to find the hypotenuse,
so 122 + 52 = c2
144 + 25 = c2
169 = c2

44. a. We have the legs a = 5, b = 12,
Pythagorean Theorem and Square Roots
Depending on which is the missing side, there are two versions
of calculation based on the Pythagorean Theorem –
finding the hypotenuse versus finding a leg.
Example B.
Find the missing side of the following right triangles.
we are to find the hypotenuse,
so 122 + 52 = c2
144 + 25 = c2
169 = c2
Hence c = 169 = 13.

45. a. We have the legs a = 5, b = 12,
Pythagorean Theorem and Square Roots
Depending on which is the missing side, there are two versions
of calculation based on the Pythagorean Theorem –
finding the hypotenuse versus finding a leg.
Example B.
Find the missing side of the following right triangles.
b. a = 5, c = 12,
we are to find the hypotenuse,
so 122 + 52 = c2
144 + 25 = c2
169 = c2
Hence c = 169 = 13.

46. a. We have the legs a = 5, b = 12,
Pythagorean Theorem and Square Roots
Depending on which is the missing side, there are two versions
of calculation based on the Pythagorean Theorem –
finding the hypotenuse versus finding a leg.
so 52 + b2 = 122
Example B.
Find the missing side of the following right triangles.
b. a = 5, c = 12, we are to find a leg,
we are to find the hypotenuse,
so 122 + 52 = c2
144 + 25 = c2
169 = c2
Hence c = 169 = 13.

47. a. We have the legs a = 5, b = 12,
Pythagorean Theorem and Square Roots
Depending on which is the missing side, there are two versions
of calculation based on the Pythagorean Theorem –
finding the hypotenuse versus finding a leg.
so 52 + b2 = 122
25 + b2 = 144
Example B.
Find the missing side of the following right triangles.
b. a = 5, c = 12, we are to find a leg,
we are to find the hypotenuse,
so 122 + 52 = c2
144 + 25 = c2
169 = c2
Hence c = 169 = 13.

48. a. We have the legs a = 5, b = 12,
Pythagorean Theorem and Square Roots
Depending on which is the missing side, there are two versions
of calculation based on the Pythagorean Theorem –
finding the hypotenuse versus finding a leg.
so 52 + b2 = 122
25 + b2 = 144
b2 = 144 – 25 = 119
Example B.
Find the missing side of the following right triangles.
b. a = 5, c = 12, we are to find a leg,
we are to find the hypotenuse,
so 122 + 52 = c2
144 + 25 = c2
169 = c2
Hence c = 169 = 13.

49. a. We have the legs a = 5, b = 12,
Pythagorean Theorem and Square Roots
Depending on which is the missing side, there are two versions
of calculation based on the Pythagorean Theorem –
finding the hypotenuse versus finding a leg.
so 52 + b2 = 122
25 + b2 = 144
b2 = 144 – 25 = 119
Hence b = 119  10.9.
Example B.
Find the missing side of the following right triangles.
b. a = 5, c = 12, we are to find a leg,
we are to find the hypotenuse,
so 122 + 52 = c2
144 + 25 = c2
169 = c2
Hence c = 169 = 13.

50. Square Rule: x2 =x x = x (all variables are > 0 below)
Rules of Radicals

51. Square Rule: x2 =x x = x (all variables are > 0 below)
Multiplication Rule: x·y = x·y
Rules of Radicals

52. Square Rule: x2 =x x = x (all variables are > 0 below)
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
Rules of Radicals

53. Square Rule: x2 =x x = x (all variables are > 0 below)
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors of the radicand to pull
out when simplifying square-root.
Rules of Radicals

54. Example A. Simplify
a. 8
Square Rule: x2 =x x = x (all variables are > 0 below)
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors of the radicand to pull
out when simplifying square-root.
Rules of Radicals

55. Example A. Simplify
a. 8 = 42
Square Rule: x2 =x x = x (all variables are > 0 below)
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors of the radicand to pull
out when simplifying square-root.
Rules of Radicals

56. Example A. Simplify
a. 8 = 42 = 22
Square Rule: x2 =x x = x (all variables are > 0 below)
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors of the radicand to pull
out when simplifying square-root.
Rules of Radicals

57. Example A. Simplify
a. 8 = 42 = 22
Square Rule: x2 =x x = x (all variables are > 0 below)
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors of the radicand to pull
out when simplifying square-root.
Rules of Radicals
b. 72 =

58. Example A. Simplify
a. 8 = 42 = 22
Square Rule: x2 =x x = x (all variables are > 0 below)
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors of the radicand to pull
out when simplifying square-root.
Rules of Radicals
b. 72 =362

59. Example A. Simplify
a. 8 = 42 = 22
Square Rule: x2 =x x = x (all variables are > 0 below)
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors of the radicand to pull
out when simplifying square-root.
Rules of Radicals
b. 72 =362 = 62

60. Example A. Simplify
a. 8 = 42 = 22
Square Rule: x2 =x x = x (all variables are > 0 below)
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors of the radicand to pull
out when simplifying square-root.
Rules of Radicals
b. 72 =362 = 62
c. x2y

61. Example A. Simplify
a. 8 = 42 = 22
Square Rule: x2 =x x = x (all variables are > 0 below)
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors of the radicand to pull
out when simplifying square-root.
Rules of Radicals
b. 72 =362 = 62
c. x2y =x2y

62. Example A. Simplify
a. 8 = 42 = 22
Square Rule: x2 =x x = x (all variables are > 0 below)
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors of the radicand to pull
out when simplifying square-root.
Rules of Radicals
b. 72 =362 = 62
c. x2y =x2y = xy

63. Example A. Simplify
a. 8 = 42 = 22
Square Rule: x2 =x x = x (all variables are > 0 below)
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors of the radicand to pull
out when simplifying square-root.
Rules of Radicals
b. 72 =362 = 62
d. x2y3
c. x2y =x2y = xy

64. Example A. Simplify
a. 8 = 42 = 22
Square Rule: x2 =x x = x (all variables are > 0 below)
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors of the radicand to pull
out when simplifying square-root.
Rules of Radicals
b. 72 =362 = 62
d. x2y3 =x2y2y
c. x2y =x2y = xy

65. Example A. Simplify
a. 8 = 42 = 22
Square Rule: x2 =x x = x (all variables are > 0 below)
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors of the radicand to pull
out when simplifying square-root.
Rules of Radicals
b. 72 =362 = 62
d. x2y3 =x2y2y = xyy
c. x2y =x2y = xy

66. Example A. Simplify
a. 8 = 42 = 22
Square Rule: x2 =x x = x (all variables are > 0 below)
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
In particular, look for square factors of the radicand to pull
out when simplifying square-root.
Rules of Radicals
b. 72 =362 = 62
d. x2y3 =x2y2y = xyy
c. x2y =x2y = xy
A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.

67. A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.
Rules of Radicals

68. A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.
Example B. Simplify.
a. 72
Rules of Radicals

69. A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.
Example B. Simplify.
a. 72 = 4 18
Rules of Radicals

70. A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.
Example B. Simplify.
a. 72 = 4 18 = 218 (not simplified yet)
Rules of Radicals

71. A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.
Example B. Simplify.
a. 72 = 4 18 = 218 (not simplified yet)
= 292
Rules of Radicals

72. A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.
Example B. Simplify.
a. 72 = 4 18 = 218 (not simplified yet)
= 292 = 2*3*2
Rules of Radicals

73. A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.
Example B. Simplify.
a. 72 = 4 18 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (simplified)
Rules of Radicals

74. A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.
Example B. Simplify.
a. 72 = 4 18 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5
Rules of Radicals

75. A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.
Example B. Simplify.
a. 72 = 4 18 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
Rules of Radicals

76. A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.
Example B. Simplify.
a. 72 = 4 18 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y
Rules of Radicals

77. A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.
Example B. Simplify.
a. 72 = 4 18 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y
Rules of Radicals
Division Rule: y
x
y
x
 =

78. A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.
Example B. Simplify.
a. 72 = 4 18 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y
Rules of Radicals
Division Rule: y
x
y
x
 =
Example C. Simplify.

79. A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.
Example B. Simplify.
a. 72 = 4 18 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y
Rules of Radicals
Division Rule: y
x
y
x
 =
Example C. Simplify.
9
4
a.

80. A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.
Example B. Simplify.
a. 72 = 4 18 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y
Rules of Radicals
Division Rule: y
x
y
x
 =
Example C. Simplify.
9
4
9
4
a. =

81. A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.
Example B. Simplify.
a. 72 = 4 18 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y
Rules of Radicals
Division Rule: y
x
y
x
 =
Example C. Simplify.
9
4
9
4
3
2
a. = =

82. A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.
Example B. Simplify.
a. 72 = 4 18 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y
Rules of Radicals
Division Rule: y
x
y
x
 =
Example C. Simplify.
9
4
9
4
3
2
9y2
x2
a. = =
b.

83. A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.
Example B. Simplify.
a. 72 = 4 18 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y
Rules of Radicals
Division Rule: y
x
y
x
 =
Example C. Simplify.
9
4
9
4
3
2
9y2
x2
9y2
x2
a. = =
b. =

84. A radical expression is said to be simplified if as much as
possible is extracted out of the square-root.
Example B. Simplify.
a. 72 = 4 18 = 218 (not simplified yet)
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y
Rules of Radicals
Division Rule: y
x
y
x
 =
Example C. Simplify.
9
4
9
4
3
2
9y2
x2
9y2
x2
3y
x
a. = =
b. = =

85. The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
Rules of Radicals

86. The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
i.e. the denominator is radical free.
Rules of Radicals

87. The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
i.e. the denominator is radical free.
If the denominator does contain radical terms, multiply the
top and bottom by suitably chosen quantities to remove the
radical term in the denominator to simplify it.
Rules of Radicals

88. Example D. Simplify
5
3
The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
i.e. the denominator is radical free.
If the denominator does contain radical terms, multiply the
top and bottom by suitably chosen quantities to remove the
radical term in the denominator to simplify it.
Rules of Radicals
a. 

89. Example D. Simplify
5
3
5·5
3·5
The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
i.e. the denominator is radical free.
If the denominator does contain radical terms, multiply the
top and bottom by suitably chosen quantities to remove the
radical term in the denominator to simplify it.
Rules of Radicals
a. = 

90. Example D. Simplify
5
3
5·5
3·5
The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
i.e. the denominator is radical free.
If the denominator does contain radical terms, multiply the
top and bottom by suitably chosen quantities to remove the
radical term in the denominator to simplify it.
Rules of Radicals
a. =  =
25
15


91. Example D. Simplify
5
3
5·5
3·5
The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
i.e. the denominator is radical free.
If the denominator does contain radical terms, multiply the
top and bottom by suitably chosen quantities to remove the
radical term in the denominator to simplify it.
Rules of Radicals
a. =  =
25
15

=
5
15

92. Example D. Simplify
5
3
5·5
3·5
The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
i.e. the denominator is radical free.
If the denominator does contain radical terms, multiply the
top and bottom by suitably chosen quantities to remove the
radical term in the denominator to simplify it.
Rules of Radicals
a. =  =
25
15

=
5
15
8x
5b. 
5
1 15or

93. Example D. Simplify
5
3
5·5
3·5
The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
i.e. the denominator is radical free.
If the denominator does contain radical terms, multiply the
top and bottom by suitably chosen quantities to remove the
radical term in the denominator to simplify it.
Rules of Radicals
a. =  =
25
15

=
5
15
8x
5
4·2x
5b. = 
5
1 15or

94. Example D. Simplify
5
3
5·5
3·5
The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
i.e. the denominator is radical free.
If the denominator does contain radical terms, multiply the
top and bottom by suitably chosen quantities to remove the
radical term in the denominator to simplify it.
Rules of Radicals
2
a. =  =
25
15

=
5
15
8x
5
4·2x
5b. =  =
2x
5

5
1 15or

95. Example D. Simplify
5
3
5·5
3·5
The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
i.e. the denominator is radical free.
If the denominator does contain radical terms, multiply the
top and bottom by suitably chosen quantities to remove the
radical term in the denominator to simplify it.
Rules of Radicals
2
a. =  =
25
15

=
5
15
8x
5
4·2x
5b. =  =
2x
5

=
2 2x
5
 2x
2x

5
1 15or

96. Example D. Simplify
5
3
5·5
3·5
The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
i.e. the denominator is radical free.
If the denominator does contain radical terms, multiply the
top and bottom by suitably chosen quantities to remove the
radical term in the denominator to simplify it.
Rules of Radicals
2
a. =  =
25
15

=
5
15
8x
5
4·2x
5b. =  =
2x
5

=
2 2x
5
 2x
2x

=
2 2x
10x
*
5
1 15or

97. Example D. Simplify
5
3
5·5
3·5
The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
i.e. the denominator is radical free.
If the denominator does contain radical terms, multiply the
top and bottom by suitably chosen quantities to remove the
radical term in the denominator to simplify it.
Rules of Radicals
2
a. =  =
25
15

=
5
15
8x
5
4·2x
5b. =  =
2x
5

=
2 2x
5
 2x
2x

=
2 2x
10x
*
=
4x
10x
5
1 15or

98. Example D. Simplify
5
3
5·5
3·5
The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
i.e. the denominator is radical free.
If the denominator does contain radical terms, multiply the
top and bottom by suitably chosen quantities to remove the
radical term in the denominator to simplify it.
Rules of Radicals
2
a. =  =
25
15

=
5
15
8x
5
4·2x
5b. =  =
2x
5

=
2 2x
5
 2x
2x

=
2 2x
10x
*
=
4x
10x
5
1 15or
4x
1 10xor

99. Example D. Simplify
5
3
5·5
3·5
The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
i.e. the denominator is radical free.
If the denominator does contain radical terms, multiply the
top and bottom by suitably chosen quantities to remove the
radical term in the denominator to simplify it.
Rules of Radicals
2
a. =  =
25
15

=
5
15
8x
5
4·2x
5b. =  =
2x
5

=
2 2x
5
 2x
2x

=
2 2x
10x
*
=
4x
10x
WARNING!!!!
a ± b = a ±b
5
1 15or
4x
1 10xor

100. Example D. Simplify
5
3
5·5
3·5
The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
i.e. the denominator is radical free.
If the denominator does contain radical terms, multiply the
top and bottom by suitably chosen quantities to remove the
radical term in the denominator to simplify it.
Rules of Radicals
2
a. =  =
25
15

=
5
15
8x
5
4·2x
5b. =  =
2x
5

=
2 2x
5
 2x
2x

=
2 2x
10x
*
=
4x
10x
WARNING!!!!
a ± b = a ±b
For example: 4 + 913 =
5
1 15or
4x
1 10xor

101. Example D. Simplify
5
3
5·5
3·5
The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
i.e. the denominator is radical free.
If the denominator does contain radical terms, multiply the
top and bottom by suitably chosen quantities to remove the
radical term in the denominator to simplify it.
Rules of Radicals
2
a. =  =
25
15

=
5
15
8x
5
4·2x
5b. =  =
2x
5

=
2 2x
5
 2x
2x

=
2 2x
10x
*
=
4x
10x
WARNING!!!!
a ± b = a ±b
For example: 4 + 913 =
5
1 15or
4x
1 10xor

102. Example D. Simplify
5
3
5·5
3·5
The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
i.e. the denominator is radical free.
If the denominator does contain radical terms, multiply the
top and bottom by suitably chosen quantities to remove the
radical term in the denominator to simplify it.
Rules of Radicals
2
a. =  =
25
15

=
5
15
8x
5
4·2x
5b. =  =
2x
5

=
2 2x
5
 2x
2x

=
2 2x
10x
*
=
4x
10x
WARNING!!!!
a ± b = a ±b
For example: 4 + 9 = 4 +913 =
5
1 15or
4x
1 10xor

103. Example D. Simplify
5
3
5·5
3·5
The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
i.e. the denominator is radical free.
If the denominator does contain radical terms, multiply the
top and bottom by suitably chosen quantities to remove the
radical term in the denominator to simplify it.
Rules of Radicals
2
a. =  =
25
15

=
5
15
8x
5
4·2x
5b. =  =
2x
5

=
2 2x
5
 2x
2x

=
2 2x
10x
*
=
4x
10x
WARNING!!!!
a ± b = a ±b
For example: 4 + 9 = 4 +9 = 2 + 3 = 513 =
5
1 15or
4x
1 10xor

104. Pythagorean Theorem and Square Roots
Rational and Irrational Numbers

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

106. The number 2 is the length of the
hypotenuse of the right triangle as shown.
Pythagorean Theorem and Square Roots
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.

107. The number 2 is the length of the
hypotenuse of the right triangle as shown.
Pythagorean Theorem and Square Roots
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.

108. The number 2 is the length of the
hypotenuse of the right triangle as shown.
Pythagorean Theorem and Square Roots
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.

109. The number 2 is the length of the
hypotenuse of the right triangle as shown.
Pythagorean Theorem and Square Roots
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.

110. The number 2 is the length of the
hypotenuse of the right triangle as shown.
Pythagorean Theorem and Square Roots
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.

111. The number 2 is the length of the
hypotenuse of the right triangle as shown.
Pythagorean Theorem and Square Roots
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?”

112. Pythagorean Theorem and Square Roots
x
3
4
Exercise C. Solve for x. Give the square–root answer and
approximate answers to the tenth place using a calculator.
1.
4
3
x2. x
12
53.
x
1
14. 2
1
x5. 6
x
6.
10
1. sqrt(0) = 2. 1 =
Exercise A. find the following square–root (no calculator).
3. 25 3. 100
5. sqrt(1/9) = 6. sqrt(1/16) = 7. sqrt(4/49)
Exercise A. Give the approximate answers to the tenth place
using a calculator.
1. sqrt(2) = 2. 3 = 3. 10 3. 0.6

113. Rules of Radicals
Exercise A. Simplify the following radicals.
1. 12 2. 18 3. 20 4. 28
5. 32 6. 36 7. 40 8. 45
9. 54 10. 60 11. 72 12. 84
13. 90 14. 96x2 15. 108x3 16. 120x2y2
17. 150y4 18. 189x3y2 19. 240x5y8 18. 242x19y34
19. 12 12 20. 1818 21. 2 16
23. 183
22. 123
24. 1227 25. 1850 26. 1040
27. 20x15x 28.12xy15y
29. 32xy324x5 30. x8y13x15y9
Exercise B. Simplify the following radicals. Remember that
you have a choice to simplify each of the radicals first then
multiply, or multiply the radicals first then simplify.

114. Rules of Radicals
Exercise C. Simplify the following radicals. Remember that
you have a choice to simplify each of the radicals first then
multiply, or multiply the radicals first then simplify. Make sure
the denominators are radical–free.
8x
531. x
10
 14
5x32. 7
20
 5
1233. 15
8x
534. 3
2
 3
32x35. 7
5
 5
236. 29
x

x
(x + 1)39. x
(x + 1)
 x
(x + 1)40. x(x + 1)
1

1
(x + 1)
37.
x
(x2 – 1)41. x(x + 1)
(x – 1)

x
(x + 1)38.
x21 – 1
Exercise D. Take the denominators of out of the radical.
42.
9x21 – 143.