Rules of Radicals

Rules of Radicals

All variables are assumed to be non-negative in the following discussion.

Square Rule: x2 =x x = x. Rules of Radicals

All variables are assumed to be non-negative in the following discussion.

Square Rule: x2 =x x = x. Multiplication Rule: x·y = x·y

Rules of Radicals

All variables are assumed to be non-negative in the following discussion.

Example A. Simplify

a. 8

Square Rule: x2 =x x = x. Multiplication Rule: x·y = x·y

We use these rules to simplify root-expressions.

Rules of Radicals

All variables are assumed to be non-negative in the following discussion.

Example A. Simplify

a. 8 =

Square Rule: x2 =x x = x. Multiplication Rule: x·y = x·y

We use these rules to simplify root-expressions. In particular, look for square factors 4, 9, 16, 25, 36,.. of the radicand to extract when simplifying square-root.

Rules of Radicals

All variables are assumed to be non-negative in the following discussion.

Example A. Simplify

a. 8 =

Square Rule: x2 =x x = x. Multiplication Rule: x·y = x·y

We use these rules to simplify root-expressions. In particular, look for square factors 4, 9, 16, 25, 36,.. of the radicand to extract when simplifying square-root.

Rules of Radicals

All variables are assumed to be non-negative in the following discussion.

The square number 4 is a factor of 8.

Example A. Simplify

a. 8 = 42 = 22

Square Rule: x2 =x x = x. Multiplication Rule: x·y = x·y

We use these rules to simplify root-expressions. In particular, look for square factors 4, 9, 16, 25, 36,.. of the radicand to extract when simplifying square-root.

Rules of Radicals

All variables are assumed to be non-negative in the following discussion.

The square number 4 is a factor of 8.

Example A. Simplify

a. 8 = 42 = 22

Square Rule: x2 =x x = x. Multiplication Rule: x·y = x·y

We use these rules to simplify root-expressions. In particular, look for square factors 4, 9, 16, 25, 36,.. of the radicand to extract when simplifying square-root.

Rules of Radicals

b. 72 =

All variables are assumed to be non-negative in the following discussion.

The square number 36is a factor of 72.

Example A. Simplify

a. 8 = 42 = 22

Square Rule: x2 =x x = x. Multiplication Rule: x·y = x·y

We use these rules to simplify root-expressions. In particular, look for square factors 4, 9, 16, 25, 36,.. of the radicand to extract when simplifying square-root.

Rules of Radicals

b. 72 =362

All variables are assumed to be non-negative in the following discussion.

The square number 36is a factor of 72.

Example A. Simplify

a. 8 = 42 = 22

Square Rule: x2 =x x = x. Multiplication Rule: x·y = x·y

We use these rules to simplify root-expressions. In particular, look for square factors 4, 9, 16, 25, 36,.. of the radicand to extract when simplifying square-root.

Rules of Radicals

b. 72 =362 = 62

All variables are assumed to be non-negative in the following discussion.

The square number 36is a factor of 72.

Example A. Simplify

a. 8 = 42 = 22

Square Rule: x2 =x x = x. Multiplication Rule: x·y = x·y

We use these rules to simplify root-expressions. In particular, look for square factors 4, 9, 16, 25, 36,.. of the radicand to extract when simplifying square-root.

Rules of Radicals

b. 72 =362 = 62

c. x2y

All variables are assumed to be non-negative in the following discussion.

Example A. Simplify

a. 8 = 42 = 22

Square Rule: x2 =x x = x. Multiplication Rule: x·y = x·y

We use these rules to simplify root-expressions. In particular, look for square factors 4, 9, 16, 25, 36,.. of the radicand to extract when simplifying square-root.

Rules of Radicals

b. 72 =362 = 62

c. x2y =x2y

All variables are assumed to be non-negative in the following discussion.

Example A. Simplify

a. 8 = 42 = 22

Square Rule: x2 =x x = x. Multiplication Rule: x·y = x·y

We use these rules to simplify root-expressions. In particular, look for square factors 4, 9, 16, 25, 36,.. of the radicand to extract when simplifying square-root.

Rules of Radicals

b. 72 =362 = 62

c. x2y =x2y = xy

All variables are assumed to be non-negative in the following discussion.

Example A. Simplify

a. 8 = 42 = 22

Square Rule: x2 =x x = x. Multiplication Rule: x·y = x·y

We use these rules to simplify root-expressions. In particular, look for square factors 4, 9, 16, 25, 36,.. of the radicand to extract when simplifying square-root.

Rules of Radicals

b. 72 =362 = 62

d. x2y3

c. x2y =x2y = xy

All variables are assumed to be non-negative in the following discussion.

Example A. Simplify

a. 8 = 42 = 22

Square Rule: x2 =x x = x. Multiplication Rule: x·y = x·y

We use these rules to simplify root-expressions. In particular, look for square factors 4, 9, 16, 25, 36,.. of the radicand to extract when simplifying square-root.

Rules of Radicals

b. 72 =362 = 62

d. x2y3 =x2y2y

c. x2y =x2y = xy

All variables are assumed to be non-negative in the following discussion.

Example A. Simplify

a. 8 = 42 = 22

Square Rule: x2 =x x = x. Multiplication Rule: x·y = x·y

We use these rules to simplify root-expressions. In particular, look for square factors 4, 9, 16, 25, 36,.. of the radicand to extract when simplifying square-root.

Rules of Radicals

b. 72 =362 = 62

d. x2y3 =x2y2y = xyy

c. x2y =x2y = xy

All variables are assumed to be non-negative in the following discussion.

Example A. Simplify

a. 8 = 42 = 22

Square Rule: x2 =x x = x. Multiplication Rule: x·y = x·y

We use these rules to simplify root-expressions. In particular, look for square factors 4, 9, 16, 25, 36,.. of the radicand to extract when simplifying square-root.

Rules of Radicals

b. 72 =362 = 62

d. x2y3 =x2y2y = xyy

c. x2y =x2y = xy

All variables are assumed to be non-negative in the following discussion.

A radical expression is said to be simplified if as much as possible is extracted out of the radical.

Example A. Simplify

a. 8 = 42 = 22

Square Rule: x2 =x x = x. Multiplication Rule: x·y = x·y

We use these rules to simplify root-expressions. In particular, look for square factors 4, 9, 16, 25, 36,.. of the radicand to extract when simplifying square-root.

Rules of Radicals

b. 72 =362 = 62

d. x2y3 =x2y2y = xyy

c. x2y =x2y = xy

All variables are assumed to be non-negative in the following discussion.

(For example 72 =98 = 38 are not simplified, but 62 is.)

A radical expression is said to be simplified if as much as possible is extracted out of the radical.

Rules of RadicalsWe may simplify a radical expression by extracting roots out of the radical in steps.

Example B. Simplify.

a. 72

Rules of RadicalsWe may simplify a radical expression by extracting roots out of the radical in steps.

Example B. Simplify.

a. 72 = 4 18

Rules of RadicalsWe may simplify a radical expression by extracting roots out of the radical in steps.

Example B. Simplify.

a. 72 = 4 18 = 218 (not simplified yet)

Rules of RadicalsWe may simplify a radical expression by extracting roots out of the radical in steps.

Example B. Simplify.

a. 72 = 4 18 = 218 (not simplified yet) = 292

Rules of RadicalsWe may simplify a radical expression by extracting roots out of the radical in steps.

We may simplify a radical expression by extracting roots out of the radical in steps. Example B. Simplify.

a. 72 = 4 18 = 218 (not simplified yet) = 292 = 2*3*2

Rules of Radicals

Example B. Simplify.

a. 72 = 4 18 = 218 (not simplified yet) = 292 = 2*3*2 = 62 (simplified)

Rules of RadicalsWe may simplify a radical expression by extracting roots out of the radical in steps.

Example B. Simplify.

a. 72 = 4 18 = 218 (not simplified yet) = 292 = 2*3*2 = 62 (simplified)b.80x4y5

Rules of RadicalsWe may simplify a radical expression by extracting roots out of the radical in steps.

Example B. Simplify.

a. 72 = 4 18 = 218 (not simplified yet) = 292 = 2*3*2 = 62 (simplified)b.80x4y5 = 16·5x4y4y

Rules of RadicalsWe may simplify a radical expression by extracting roots out of the radical in steps.

Example B. Simplify.

a. 72 = 4 18 = 218 (not simplified yet) = 292 = 2*3*2 = 62 (simplified)b.80x4y5 = 16·5x4y4y = 4x2y25y

Rules of RadicalsWe may simplify a radical expression by extracting roots out of the radical in steps.

Example B. Simplify.

a. 72 = 4 18 = 218 (not simplified yet) = 292 = 2*3*2 = 62 (simplified)b.80x4y5 = 16·5x4y4y = 4x2y25y

Rules of Radicals

Division Rule: yx

yx =

We may simplify a radical expression by extracting roots out of the radical in steps.

Example B. Simplify.

a. 72 = 4 18 = 218 (not simplified yet) = 292 = 2*3*2 = 62 (simplified)b.80x4y5 = 16·5x4y4y = 4x2y25y

Rules of Radicals

Division Rule: yx

yx =

Example C. Simplify.

We may simplify a radical expression by extracting roots out of the radical in steps.

Example B. Simplify.

a. 72 = 4 18 = 218 (not simplified yet) = 292 = 2*3*2 = 62 (simplified)b.80x4y5 = 16·5x4y4y = 4x2y25y

Rules of Radicals

Division Rule: yx

yx =

Example C. Simplify.

94a.

We may simplify a radical expression by extracting roots out of the radical in steps.

Example B. Simplify.

a. 72 = 4 18 = 218 (not simplified yet) = 292 = 2*3*2 = 62 (simplified)b.80x4y5 = 16·5x4y4y = 4x2y25y

Rules of Radicals

Division Rule: yx

yx =

Example C. Simplify.

94

94a. =

We may simplify a radical expression by extracting roots out of the radical in steps.

Example B. Simplify.

a. 72 = 4 18 = 218 (not simplified yet) = 292 = 2*3*2 = 62 (simplified)b.80x4y5 = 16·5x4y4y = 4x2y25y

Rules of Radicals

Division Rule: yx

yx =

Example C. Simplify.

94

94

32a. = =

We may simplify a radical expression by extracting roots out of the radical in steps.

Example B. Simplify.

a. 72 = 4 18 = 218 (not simplified yet) = 292 = 2*3*2 = 62 (simplified)b.80x4y5 = 16·5x4y4y = 4x2y25y

Rules of Radicals

Division Rule: yx

yx =

Example C. Simplify.

94

94

32

9y2x2

a. = =

b.

We may simplify a radical expression by extracting roots out of the radical in steps.

Example B. Simplify.

a. 72 = 4 18 = 218 (not simplified yet) = 292 = 2*3*2 = 62 (simplified)b.80x4y5 = 16·5x4y4y = 4x2y25y

Rules of Radicals

Division Rule: yx

yx =

Example C. Simplify.

94

94

32

9y2x2

9y2

x2

a. = =

b. =

We may simplify a radical expression by extracting roots out of the radical in steps.

Example B. Simplify.

a. 72 = 4 18 = 218 (not simplified yet) = 292 = 2*3*2 = 62 (simplified)b.80x4y5 = 16·5x4y4y = 4x2y25y

Rules of Radicals

Division Rule: yx

yx =

Example C. Simplify.

94

94

32

9y2x2

9y2

x2

3yx

a. = =

b. = =

We may simplify a radical expression by extracting roots out of the radical in steps.

The root of a fraction is said to be simplified if the denominator does not have any root-terms.

Rules of Radicals

Example D. Simplify

53a.

The root of a fraction is said to be simplified if the denominator does not have any root-terms. If not, the denominator needs to be changed to be root free.

Rules of Radicals

Example D. Simplify

53a.

The root of a fraction is said to be simplified if the denominator does not have any root-terms. If not, the denominator needs to be changed to be root 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.

Rules of Radicals

Example D. Simplify

53a.

Example D. Simplify

53

5·53·5

The root of a fraction is said to be simplified if the denominator does not have any root-terms. If not, the denominator needs to be changed to be root 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.

Rules of Radicals

a. =

Multiple an extra 5 to extract root to make denominator to be root-free.

Example D. Simplify

53

5·53·5

The root of a fraction is said to be simplified if the denominator does not have any root-terms. If not, the denominator needs to be changed to be root 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.

Rules of Radicals

a. = =2515

Multiple an extra 5 to extract root to make denominator to be root-free.

Example D. Simplify

53

5·53·5

The root of a fraction is said to be simplified if the denominator does not have any root-terms. If not, the denominator needs to be changed to be root 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.

Rules of Radicals

a. = =2515

= 5

15

Multiple an extra 5 to extract root to make denominator to be root-free.

Simplified! Became the denominator is root-free (and the numerator is simplified).

Example D. Simplify

53

5·53·5

The root of a fraction is said to be simplified if the denominator does not have any root-terms. If not, the denominator needs to be changed to be root 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.

Rules of Radicals

a. = =2515

= 5

15

Multiple an extra 5 to extract root to make denominator to be root-free.

Simplified! Became the denominator is root-free (and the numerator is simplified).

51 15or

Example D. Simplify

53

5·53·5

The root of a fraction is said to be simplified if the denominator does not have any root-terms. If not, the denominator needs to be changed to be root 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.

Rules of Radicals

a. = =2515

= 5

15

8x5b.

51 15or

Example D. Simplify

53

5·53·5

The root of a fraction is said to be simplified if the denominator does not have any root-terms. If not, the denominator needs to be changed to be root 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.

Rules of Radicals

a. = =2515

= 5

15

8x5

4·2x5b. =

51 15or

Example D. Simplify

53

5·53·5

The root of a fraction is said to be simplified if the denominator does not have any root-terms. If not, the denominator needs to be changed to be root 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.

Rules of Radicals

2

a. = =2515

= 5

15

8x5

4·2x5b. = =

2x5

51 15or

Example D. Simplify

53

5·53·5

The root of a fraction is said to be simplified if the denominator does not have any root-terms. If not, the denominator needs to be changed to be root 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.

Rules of Radicals

2

a. = =2515

= 5

15

8x5

4·2x5b. = =

2x5

=

2 2x5

2x2x

51 15or

Example D. Simplify

53

5·53·5

The root of a fraction is said to be simplified if the denominator does not have any root-terms. If not, the denominator needs to be changed to be root 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.

Rules of Radicals

2

a. = =2515

= 5

15

8x5

4·2x5b. = =

2x5

=

2 2x5

2x2x

= 2 2x

10x*

51 15or

Example D. Simplify

53

5·53·5

The root of a fraction is said to be simplified if the denominator does not have any root-terms. If not, the denominator needs to be changed to be root 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.

Rules of Radicals

2

a. = =2515

= 5

15

8x5

4·2x5b. = =

2x5

=

2 2x5

2x2x

= 2 2x

10x*

= 4x10x

51 15or

Example D. Simplify

53

5·53·5

The root of a fraction is said to be simplified if the denominator does not have any root-terms. If not, the denominator needs to be changed to be root 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.

Rules of Radicals

2

a. = =2515

= 5

15

8x5

4·2x5b. = =

2x5

=

2 2x5

2x2x

= 2 2x

10x*

= 4x10x

51 15or

4x1 10xor

Example D. Simplify

53

5·53·5

The root of a fraction is said to be simplified if the denominator does not have any root-terms. If not, the denominator needs to be changed to be root 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.

Rules of Radicals

2

a. = =2515

= 5

15

8x5

4·2x5b. = =

2x5

=

2 2x5

2x2x

= 2 2x

10x*

= 4x10x

WARNING!!!!a ± b = a ±b

51 15or

4x1 10xor

Example D. Simplify

53

5·53·5

The root of a fraction is said to be simplified if the denominator does not have any root-terms. If not, the denominator needs to be changed to be root 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.

Rules of Radicals

2

a. = =2515

= 5

15

8x5

4·2x5b. = =

2x5

=

2 2x5

2x2x

= 2 2x

10x*

= 4x10x

WARNING!!!!a ± b = a ±bFor example: 4 + 913 =

51 15or

4x1 10xor

Example D. Simplify

53

5·53·5

The root of a fraction is said to be simplified if the denominator does not have any root-terms. If not, the denominator needs to be changed to be root 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.

Rules of Radicals

2

a. = =2515

= 5

15

8x5

4·2x5b. = =

2x5

=

2 2x5

2x2x

= 2 2x

10x*

= 4x10x

WARNING!!!!a ± b = a ±bFor example: 4 + 913 =

51 15or

4x1 10xor

Example D. Simplify

53

5·53·5

The root of a fraction is said to be simplified if the denominator does not have any root-terms. If not, the denominator needs to be changed to be root 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.

Rules of Radicals

2

a. = =2515

= 5

15

8x5

4·2x5b. = =

2x5

=

2 2x5

2x2x

= 2 2x

10x*

= 4x10x

WARNING!!!!a ± b = a ±bFor example: 4 + 9 = 4 +913 =

51 15or

4x1 10xor

Example D. Simplify

53

5·53·5

The root of a fraction is said to be simplified if the denominator does not have any root-terms. If not, the denominator needs to be changed to be root 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.

Rules of Radicals

2

a. = =2515

= 5

15

8x5

4·2x5b. = =

2x5

=

2 2x5

2x2x

= 2 2x

10x*

= 4x10x

WARNING!!!!a ± b = a ±bFor example: 4 + 9 = 4 +9 = 2 + 3 = 513 =

51 15or

4x1 10xor

Rules of RadicalsExercise 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. 8413. 90 14. 96x2 15. 108x3 16. 120x2y2

17. 150y4 18. 189x3y2 19. 240x5y8 18. 242x19y34

19. 12 12 20. 1818 21. 2 16

23. 183

22. 123

24. 1227 25. 1850 26. 1040

27. 20x15x 28.12xy15y29. 32xy324x5 30. x8y13x15y9

Exercise B. Simplify the following radicals. Remember thatyou have a choice to simplify each of the radicals first then multiply, or multiply the radicals first then simplify.

Rules of RadicalsExercise C. Simplify the following radicals. Remember thatyou 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.

8x531. x

10 145x32. 7

20 51233. 15

8x534. 3

2 332x35. 7

5 5236. 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 – 1Exercise D. Take the denominators of out of the radical.

42. 9x21 – 143.