4 2 rules of radicals

Rules of Radicals
Frank Ma © 2011

Rules of Radicals
Square Rule: x2 =x x= x if x > 0.

Rules of Radicals
Multiplication Rule: x·y = x·y

Rules of Radicals
We use these rules to simplify root-expressions.

Rules of Radicals
In particular, look for square factors of the radicand to pull out when simplifying square-root.

Rules of Radicals
Example A. Simplify
8

Rules of Radicals
Example A. Simplify
8 = 42

Rules of Radicals
Example A. Simplify
8 = 42 = 22

Rules of Radicals
Example A. Simplify
8 = 42 = 22
b. 72 =

Rules of Radicals
Example A. Simplify
8 = 42 = 22
b. 72 =362

Rules of Radicals
Example A. Simplify
8 = 42 = 22
b. 72 =362 = 62

Rules of Radicals
Example A. Simplify
8 = 42 = 22
b. 72 =362 = 62
c. x2y

Rules of Radicals
Example A. Simplify
8 = 42 = 22
b. 72 =362 = 62
c. x2y =x2y

Rules of Radicals
Example A. Simplify
8 = 42 = 22
b. 72 =362 = 62
c. x2y =x2y = xy

Rules of Radicals
Example A. Simplify
8 = 42 = 22
b. 72 =362 = 62
c. x2y =x2y = xy
d. x2y3

Rules of Radicals
Example A. Simplify
8 = 42 = 22
b. 72 =362 = 62
c. x2y =x2y = xy
d. x2y3 =x2y2y

Rules of Radicals
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

Rules of Radicals
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.

Rules of Radicals

Rules of Radicals
Example B. Simplify.
a. 72

Rules of Radicals
a. 72 = 418

Rules of Radicals
a. 72 = 418 = 218 (not simplified yet)

Rules of Radicals
= 292

Rules of Radicals
= 292 = 2*3*2

Rules of Radicals
= 292 = 2*3*2 = 62 (simplified)

Rules of Radicals
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5

Rules of Radicals
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y

Rules of Radicals
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y

Rules of Radicals
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y
x
x

=
Division Rule:
y
y

Rules of Radicals
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y
x
x

=
Division Rule:
y
y
Example C. Simplify.

Rules of Radicals
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y
x
x

=
Division Rule:
y
y
4

a.
9

Rules of Radicals
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y
x
x

=
Division Rule:
y
y
4
4

a.
=
9
9

Rules of Radicals
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y
x
x

=
Division Rule:
y
y
2
4
4

a.
=
=
3
9
9

Rules of Radicals
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y
x
x

=
Division Rule:
y
y
2
4
4

a.
=
=
3
9
9
x2

b.
9y2

Rules of Radicals
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y
x
x

=
Division Rule:
y
y
2
4
4

a.
=
=
3
9
9
x2
x2

b.
=
9y2
9y2

Rules of Radicals
= 292 = 2*3*2 = 62 (simplified)
b.80x4y5 = 16·5x4y4y
= 4x2y25y
x
x

=
Division Rule:
y
y
2
4
4

a.
=
=
3
9
9
x2
x2

x
b.
=
=
9y2
9y2
3y

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

Rules of Radicals
if the denominator is completely extracted out of the radical, i.e. the denominator is radical free.

Rules of Radicals
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
Example D. Simplify
3

a.
5

Rules of Radicals
Example D. Simplify
3
3·5


a.
=
5
5·5

Rules of Radicals
Example D. Simplify
3
3·5
15



a.
=
=
5
5·5
25


Rules of Radicals
Example D. Simplify
3
3·5
15
15




a.
=
=
=
5
5·5
5
25


Rules of Radicals
Example D. Simplify
3
3·5
15
15
1




a.
=
15
=
=
or
5
5·5
5
5
25

5

b.
8x

Rules of Radicals
Example D. Simplify
3
3·5
15
15
1




a.
=
15
=
=
or
5
5·5
5
5
25

5
5


b.
=
8x
4·2x

Rules of Radicals
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


Rules of Radicals
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


Rules of Radicals
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

*

Rules of Radicals
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

*

Rules of Radicals
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

Rules of Radicals
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

Rules of Radicals
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
=

Rules of Radicals
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
=

Rules of Radicals
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
=

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
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

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.
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

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