# 4 2 rules of radicals

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## 4 2 rules of radicalsPresentation Transcript

Square Rule: x2 =x x= x if x > 0.
Square Rule: x2 =x x= x if x > 0.
Multiplication Rule: x·y = x·y
Square Rule: x2 =x x= x if x > 0.
Multiplication Rule: x·y = x·y
We use these rules to simplify root-expressions.
Square Rule: x2 =x x= x if x > 0.
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.
Square Rule: x2 =x x= x if x > 0.
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.
Example A. Simplify
8
Square Rule: x2 =x x= x if x > 0.
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.
Example A. Simplify
8 = 42
Square Rule: x2 =x x= x if x > 0.
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.
Example A. Simplify
8 = 42 = 22
Square Rule: x2 =x x= x if x > 0.
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.
Example A. Simplify
8 = 42 = 22
b. 72 =
Square Rule: x2 =x x= x if x > 0.
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.
Example A. Simplify
8 = 42 = 22
b. 72 =362
Square Rule: x2 =x x= x if x > 0.
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.
Example A. Simplify
8 = 42 = 22
b. 72 =362 = 62
Square Rule: x2 =x x= x if x > 0.
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.
Example A. Simplify
8 = 42 = 22
b. 72 =362 = 62
c. x2y
Square Rule: x2 =x x= x if x > 0.
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.
Example A. Simplify
8 = 42 = 22
b. 72 =362 = 62
c. x2y =x2y
Square Rule: x2 =x x= x if x > 0.
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.
Example A. Simplify
8 = 42 = 22
b. 72 =362 = 62
c. x2y =x2y = xy
Square Rule: x2 =x x= x if x > 0.
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.
Example A. Simplify
8 = 42 = 22
b. 72 =362 = 62
c. x2y =x2y = xy
d. x2y3
Square Rule: x2 =x x= x if x > 0.
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.
Example A. Simplify
8 = 42 = 22
b. 72 =362 = 62
c. x2y =x2y = xy
d. x2y3 =x2y2y
Square Rule: x2 =x x= x if x > 0.
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.
Example A. Simplify
8 = 42 = 22
b. 72 =362 = 62
c. x2y =x2y = xy
d. x2y3 =x2y2y = xyy
Square Rule: x2 =x x= x if x > 0.
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.
Example A. Simplify
8 = 42 = 22
b. 72 =362 = 62
c. x2y =x2y = xy
d. x2y3 =x2y2y = xyy
A radical expression is said to be simplifiedif as much as possible is extracted out of the square-root.
A radical expression is said to be simplifiedif as much as possible is extracted out of the square-root.
A radical expression is said to be simplifiedif as much as possible is extracted out of the square-root.
Example B. Simplify.
a. 72
A radical expression is said to be simplifiedif as much as possible is extracted out of the square-root.
Example B. Simplify.
a. 72 = 418
A radical expression is said to be simplifiedif as much as possible is extracted out of the square-root.
Example B. Simplify.
a. 72 = 418 = 218 (not simplified yet)
A radical expression is said to be simplifiedif 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
A radical expression is said to be simplifiedif 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
A radical expression is said to be simplifiedif 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)
A radical expression is said to be simplifiedif 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
A radical expression is said to be simplifiedif 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
A radical expression is said to be simplifiedif 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
A radical expression is said to be simplifiedif 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
x
x

=
Division Rule:
y
y
A radical expression is said to be simplifiedif 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
x
x

=
Division Rule:
y
y
Example C. Simplify.
A radical expression is said to be simplifiedif 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
x
x

=
Division Rule:
y
y
Example C. Simplify.
4

a.
9
A radical expression is said to be simplifiedif 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
x
x

=
Division Rule:
y
y
Example C. Simplify.
4
4

a.
=
9
9
A radical expression is said to be simplifiedif 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
x
x

=
Division Rule:
y
y
Example C. Simplify.
2
4
4

a.
=
=
3
9
9
A radical expression is said to be simplifiedif 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
x
x

=
Division Rule:
y
y
Example C. Simplify.
2
4
4

a.
=
=
3
9
9
x2

b.
9y2
A radical expression is said to be simplifiedif 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
x
x

=
Division Rule:
y
y
Example C. Simplify.
2
4
4

a.
=
=
3
9
9
x2
x2

b.
=
9y2
9y2
A radical expression is said to be simplifiedif 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
x
x

=
Division Rule:
y
y
Example C. Simplify.
2
4
4

a.
=
=
3
9
9
x2
x2

x
b.
=
=
9y2
9y2
3y
The radical of a fractional expression is said to be simplified
if the denominator is completely extracted out of the radical,
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.
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.
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.
Example D. Simplify
3

a.
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.
Example D. Simplify
3
3·5

a.
=
5
5·5
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.
Example D. Simplify
3
3·5
15

a.
=
=
5
5·5
25

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.
Example D. Simplify
3
3·5
15
15

a.
=
=
=
5
5·5
5
25

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.
Example D. Simplify
3
3·5
15
15
1

a.
=
15
=
=
or
5
5·5
5
5
25

5

b.
8x
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.
Example D. Simplify
3
3·5
15
15
1

a.
=
15
=
=
or
5
5·5
5
5
25

5
5

b.
=
8x
4·2x
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.
Example D. Simplify
3
3·5
15
15
1

a.
=
15
=
=
or
5
5·5
5
5
25

5
5
5

b.
=
=
8x
4·2x
2
2x

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.
Example D. Simplify
3
3·5
15
15
1

a.
=
15
=
=
or
5
5·5
5
5
25

5
5
5
5
2x

b.
=
=
=
8x
4·2x
2
2x

2
2x

2x

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.
Example D. Simplify
3
3·5
15
15
1

a.
=
15
=
=
or
5
5·5
5
5
25

5
5
5
5
2x
10x

b.
=
=
=
=
8x
4·2x
2
2x
2
2x

2
2x

2x

*
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.
Example D. Simplify
3
3·5
15
15
1

a.
=
15
=
=
or
5
5·5
5
5
25

5
5
5
5
2x
10x
10x

b.
=
=
=
=
=
8x
4·2x
2
2x
4x
2
2x

2
2x

2x

*
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.
Example D. Simplify
3
3·5
15
15
1

a.
=
15
=
=
or
5
5·5
5
5
25

5
5
5
5
2x
10x
10x

b.
=
=
=
=
=
8x
4·2x
2
2x
4x
2
2x

2
2x

2x

*
1
10x
or
4x
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.
Example D. Simplify
3
3·5
15
15
1

a.
=
15
=
=
or
5
5·5
5
5
25

5
5
5
5
2x
10x
10x

b.
=
=
=
=
=
8x
4·2x
2
2x
4x
2
2x

2
2x

2x

*
1
WARNING!!!!
10x
or
4x
a ± b = a ±b
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.
Example D. Simplify
3
3·5
15
15
1

a.
=
15
=
=
or
5
5·5
5
5
25

5
5
5
5
2x
10x
10x

b.
=
=
=
=
=
8x
4·2x
2
2x
4x
2
2x

2
2x

2x

*
1
WARNING!!!!
10x
or
4x
a ± b = a ±b
For example:
4 + 9
13
=
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.
Example D. Simplify
3
3·5
15
15
1

a.
=
15
=
=
or
5
5·5
5
5
25

5
5
5
5
2x
10x
10x

b.
=
=
=
=
=
8x
4·2x
2
2x
4x
2
2x

2
2x

2x

*
1
WARNING!!!!
10x
or
4x
a ± b = a ±b
For example:
4 + 9
13
=
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.
Example D. Simplify
3
3·5
15
15
1

a.
=
15
=
=
or
5
5·5
5
5
25

5
5
5
5
2x
10x
10x

b.
=
=
=
=
=
8x
4·2x
2
2x
4x
2
2x

2
2x

2x

*
1
WARNING!!!!
10x
or
4x
a ± b = a ±b
For example:
4 + 9 = 4 +9
13
=
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.
Example D. Simplify
3
3·5
15
15
1

a.
=
15
=
=
or
5
5·5
5
5
25

5
5
5
5
2x
10x
10x

b.
=
=
=
=
=
8x
4·2x
2
2x
4x
2
2x

2
2x

2x

*
1
WARNING!!!!
10x
or
4x
a ± b = a ±b
For example:
4 + 9 = 4 +9 = 2 + 3 = 5
13
=
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
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.
19. 12 12
20. 1818
21. 2 16
22. 123
23. 183
24. 1227
25. 1850
26. 1040
28.12xy15y
27. 20x15x
30. x8y13x15y9
29. 32xy324x5
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.
5
10
5x
20
12
15

31.
32.
33.
8x
x
14
7
5
5
2
32x
5
2
x

34.
35.
36.
8x
3
3
7
5
29
1

(x + 1)

37.
38.
(x + 1)
x
(x + 1)
1

(x + 1)
(x + 1)

40.
39.
x
x(x + 1)
x
x
(x2 – 1)
(x – 1)

41.
x
x(x + 1)
Exercise D. Take the denominators of out of the radical.
1
1

42.
43.
1 –
1 –
x2
9x2