4 4 more on algebra of radicals-x

More on Algebra of Radicals

Remember that x·y = x·y, x·x = xMore on Algebra of Radicals

yx

yx =,

Remember that x·y = x·y,

3*3 *2* 2 *2 *3 *2

x·x = xMore on Algebra of Radicals

Example A. Simplifya.

yx

yx =,


3*3 *2* 2 *2 *3 *2

= 3 * 2 *3 *3 * 2 *2 *2

x·x = xMore on Algebra of Radicals

Example A. Simplifya.

yx

yx =,


3*3 *2* 2 *2 *3 *2

= 3 * 2 *3 *3 * 2 *2 *2

x·x = x

3


Example A. Simplify

2a.

yx

yx =,


3*3 *2* 2 *2 *3 *2

= 3 * 2 *3 *3 * 2 *2 *2= 3 * 3 * 2 * 2 * 2

x·x = x

3


Example A. Simplify

2a.

yx

yx =,


3*3 *2* 2 *2 *3 *2

= 3 * 2 *3 *3 * 2 *2 *2= 3 * 3 * 2 * 2 * 2 = 362

x·x = x

3


Example A. Simplify

2a.

yx

yx =,


3*3 *2* 2 *2 *3 *2

= 3 * 2 *3 *3 * 2 *2 *2= 3 * 3 * 2 * 2 * 2 = 362

x·x = x

3


Example A. Simplify

2

b. 12 (3 + 32)

a.

yx

yx =,


3*3 *2* 2 *2 *3 *2

= 3 * 2 *3 *3 * 2 *2 *2= 3 * 3 * 2 * 2 * 2 = 362

x·x = x

3


Example A. Simplify

2

b. 12 (3 + 32) = 123 + 3122

a.

yx

yx =,


3*3 *2* 2 *2 *3 *2

= 3 * 2 *3 *3 * 2 *2 *2= 3 * 3 * 2 * 2 * 2 = 362

x·x = x

3


Example A. Simplify

2

b. 12 (3 + 32) = 123 + 3122 = 36+ 324

a.

yx

yx =,


3*3 *2* 2 *2 *3 *2

= 3 * 2 *3 *3 * 2 *2 *2= 3 * 3 * 2 * 2 * 2 = 362

x·x = x

3


Example A. Simplify

2

b. 12 (3 + 32) = 123 + 3122 = 36+ 324 = 6 + 3√4*6

a.

yx

yx =,


3*3 *2* 2 *2 *3 *2

= 3 * 2 *3 *3 * 2 *2 *2= 3 * 3 * 2 * 2 * 2 = 362

x·x = x

3


Example A. Simplify

2

b. 12 (3 + 32) = 123 + 3122 = 36+ 324 = 6 + 3√4*6 = 6 + 3*26

a.

yx

yx =,


3*3 *2* 2 *2 *3 *2

= 3 * 2 *3 *3 * 2 *2 *2= 3 * 3 * 2 * 2 * 2 = 362

x·x = x

3


Example A. Simplify

2

b. 12 (3 + 32) = 123 + 3122 = 36+ 324 = 6 + 3√4*6 = 6 + 3*26 = 6 + 66

a.

yx

yx =,


3*3 *2* 2 *2 *3 *2

= 3 * 2 *3 *3 * 2 *2 *2= 3 * 3 * 2 * 2 * 2 = 362

x·x = x

3


Example A. Simplify

2

b. 12 (3 + 32) = 123 + 3122 = 36+ 324 = 6 + 3√4*6 = 6 + 3*26 = 6 + 66

a.

(Remember 6 + 6√6 = 12√6 because they are not like-terms.)

yx

yx =,

c. (33 – 22)(23 + 32) More on Algebra of Radicals

c. (33 – 22)(23 + 32) = 33*23


c. (33 – 22)(23 + 32) = 33*23 + 33*32


c. (33 – 22)(23 + 32) = 33*23 + 33*32 – 22*23


c. (33 – 22)(23 + 32) = 33*23 + 33*32 – 22*23 – 22*32


c. (33 – 22)(23 + 32) = 33*23 + 33*32 – 22*23 – 22*32


3

c. (33 – 22)(23 + 32) = 33*23 + 33*32 – 22*23 – 22*32


3 √6 √6

c. (33 – 22)(23 + 32) = 33*23 + 33*32 – 22*23 – 22*32


3 2√6 √6

c. (33 – 22)(23 + 32) = 33*23 + 33*32 – 22*23 – 22*32


= 18 + 96 – 46 – 123 2√6 √6

c. (33 – 22)(23 + 32) = 33*23 + 33*32 – 22*23 – 22*32


= 18 + 96 – 46 – 12 = 6 + 56

3 2√6 √6

c. (33 – 22)(23 + 32) = 33*23 + 33*32 – 22*23 – 22*32


= 18 + 96 – 46 – 12 = 6 + 56

3 2√6 √6

Conjugates

c. (33 – 22)(23 + 32) = 33*23 + 33*32 – 22*23 – 22*32


= 18 + 96 – 46 – 12 = 6 + 56

3 2√6 √6

Conjugates We call x + y, x – y the conjugate of each other.

c. (33 – 22)(23 + 32) = 33*23 + 33*32 – 22*23 – 22*32


= 18 + 96 – 46 – 12 = 6 + 56

3 2√6 √6

Conjugates We call x + y, x – y the conjugate of each other. For example, the conjugate of 3 – 25 is 3 + 25,

c. (33 – 22)(23 + 32) = 33*23 + 33*32 – 22*23 – 22*32


= 18 + 96 – 46 – 12 = 6 + 56

3 2√6 √6

Conjugates We call x + y, x – y the conjugate of each other. For example, the conjugate of 3 – 25 is 3 + 25,the conjugate of 5 + 22 is 5 – 22.

c. (33 – 22)(23 + 32) = 33*23 + 33*32 – 22*23 – 22*32


= 18 + 96 – 46 – 12 = 6 + 56

3 2√6 √6

Conjugates We call x + y, x – y the conjugate of each other. For example, the conjugate of 3 – 25 is 3 + 25,the conjugate of 5 + 22 is 5 – 22.The importance of conjugate pair is that (x + y)(x – y) = x2 – y2

c. (33 – 22)(23 + 32) = 33*23 + 33*32 – 22*23 – 22*32


= 18 + 96 – 46 – 12 = 6 + 56

3 2√6 √6

Conjugates We call x + y, x – y the conjugate of each other. For example, the conjugate of 3 – 25 is 3 + 25,the conjugate of 5 + 22 is 5 – 22.The importance of conjugate pair is that (x + y)(x – y) = x2 – y2 Example B. Multiply the following conjugates.a. (3 – 25)(3 + 25)

c. (33 – 22)(23 + 32) = 33*23 + 33*32 – 22*23 – 22*32


= 18 + 96 – 46 – 12 = 6 + 56

3 2√6 √6

Conjugates We call x + y, x – y the conjugate of each other. For example, the conjugate of 3 – 25 is 3 + 25,the conjugate of 5 + 22 is 5 – 22.The importance of conjugate pair is that (x + y)(x – y) = x2 – y2 Example B. Multiply the following conjugates.a. (3 – 25)(3 + 25) = 32 – (25)2

c. (33 – 22)(23 + 32) = 33*23 + 33*32 – 22*23 – 22*32


= 18 + 96 – 46 – 12 = 6 + 56

3 2√6 √6

Conjugates We call x + y, x – y the conjugate of each other. For example, the conjugate of 3 – 25 is 3 + 25,the conjugate of 5 + 22 is 5 – 22.The importance of conjugate pair is that (x + y)(x – y) = x2 – y2 Example B. Multiply the following conjugates.a. (3 – 25)(3 + 25) = 32 – (25)2

= 9 – 4*5

b.(5 + 22)(3 – 7)(5 – 22)(3 + 7)More on Algebra of Radicals

b.(5 + 22)(3 – 7)(5 – 22)(3 + 7) =(5 + 22)(5 – 22)(3 – 7)(3 + 7)


b.(5 + 22)(3 – 7)(5 – 22)(3 + 7) =(5 + 22)(5 – 22)(3 – 7)(3 + 7) = ((5)2 – (22)2) (32 – (7)2)


b.(5 + 22)(3 – 7)(5 – 22)(3 + 7) =(5 + 22)(5 – 22)(3 – 7)(3 + 7) = ((5)2 – (22)2) (32 – (7)2) = (5 – 8)(9 – 7)


b.(5 + 22)(3 – 7)(5 – 22)(3 + 7) =(5 + 22)(5 – 22)(3 – 7)(3 + 7) = ((5)2 – (22)2) (32 – (7)2) = (5 – 8)(9 – 7) = –3*2 = –6


b.(5 + 22)(3 – 7)(5 – 22)(3 + 7) =(5 + 22)(5 – 22)(3 – 7)(3 + 7) = ((5)2 – (22)2) (32 – (7)2) = (5 – 8)(9 – 7) = –3*2 = –6


Conjugates are used to rationalize the denominator, i.e. to rewrite a fraction with square-root term(s) in the denominator so it does not contain any radical.

b.(5 + 22)(3 – 7)(5 – 22)(3 + 7) =(5 + 22)(5 – 22)(3 – 7)(3 + 7) = ((5)2 – (22)2) (32 – (7)2) = (5 – 8)(9 – 7) = –3*2 = –6


Conjugates are used to rationalize the denominator, i.e. to rewrite a fraction with square-root term(s) in the denominator so it does not contain any radical. Example C. Rationalize the following expressions.

2 2 – 35

a.

b.(5 + 22)(3 – 7)(5 – 22)(3 + 7) =(5 + 22)(5 – 22)(3 – 7)(3 + 7) = ((5)2 – (22)2) (32 – (7)2) = (5 – 8)(9 – 7) = –3*2 = –6



2 2 – 35

a. Multiply the top and bottom by the conjugate of the denominator.

b.(5 + 22)(3 – 7)(5 – 22)(3 + 7) =(5 + 22)(5 – 22)(3 – 7)(3 + 7) = ((5)2 – (22)2) (32 – (7)2) = (5 – 8)(9 – 7) = –3*2 = –6



2 2 – 35

a.

= 2 (2 – 35)

Multiply the top and bottom by the conjugate of the denominator.

b.(5 + 22)(3 – 7)(5 – 22)(3 + 7) =(5 + 22)(5 – 22)(3 – 7)(3 + 7) = ((5)2 – (22)2) (32 – (7)2) = (5 – 8)(9 – 7) = –3*2 = –6



2 2 – 35

a.

= 2 (2 – 35)

(2 + 35) (2 + 35)


b.(5 + 22)(3 – 7)(5 – 22)(3 + 7) =(5 + 22)(5 – 22)(3 – 7)(3 + 7) = ((5)2 – (22)2) (32 – (7)2) = (5 – 8)(9 – 7) = –3*2 = –6



2 2 – 35

a.

= 2 (2 – 35)

(2 + 35) (2 + 35)


(2)2 – (35)2 = 4 – 45 = –41

b.(5 + 22)(3 – 7)(5 – 22)(3 + 7) =(5 + 22)(5 – 22)(3 – 7)(3 + 7) = ((5)2 – (22)2) (32 – (7)2) = (5 – 8)(9 – 7) = –3*2 = –6



2 2 – 35

a.

= 2 (2 – 35)

(2 + 35) (2 + 35)


= 4 + 6√5– 41

(2)2 – (35)2 = 4 – 45 = –41

b. 5 – 23 3 + 43


b. 5 – 23 3 + 43

More on Algebra of RadicalsMultiply the top and bottom by the conjugate of the denominator.

b. 5 – 23 3 + 43

= (3 – 43) (3 – 43)·


(5 – 23) (3 + 43)


b. 5 – 23 3 + 43

= (3 – 43) (3 – 43)·


(5 – 23) (3 + 43)


(3)2 – (43)2

b. 5 – 23 3 + 43

= (3 – 43) (3 – 43)·


(5 – 23) (3 + 43)


(3)2 – (43)2 = 9 – 48 = –39

b. 5 – 23 3 + 43

= (3 – 43) (3 – 43)·

(5 – 23) (3 – 43)


(5 – 23) (3 + 43)


(3)2 – (43)2 = 9 – 48 = –39

= –39

b. 5 – 23 3 + 43

= (3 – 43) (3 – 43)·

(5 – 23) (3 – 43)

= –39


(5 – 23) (3 + 43)


(3)2 – (43)2 = 9 – 48 = –39

= –3915 – 63 – 203 + 24

b. 5 – 23 3 + 43

= (3 – 43) (3 – 43)·

(5 – 23) (3 – 43)

= –39

= 39 – 263–39


(5 – 23) (3 + 43)


(3)2 – (43)2 = 9 – 48 = –39

= –3915 – 63 – 203 + 24

b. 5 – 23 3 + 43

= (3 – 43) (3 – 43)·

(5 – 23) (3 – 43)

= –39

= 39 – 263–39


(5 – 23) (3 + 43)


(3)2 – (43)2 = 9 – 48 = –39

= –3915 – 63 – 203 + 24

= 13(3 – 23)–39

b. 5 – 23 3 + 43

= (3 – 43) (3 – 43)·

(5 – 23) (3 – 43)

= –39

= 39 – 263–39

–1


(5 – 23) (3 + 43)


(3)2 – (43)2 = 9 – 48 = –39

= –3915 – 63 – 203 + 24

= 13(3 – 23)–393

b. 5 – 23 3 + 43

= (3 – 43) (3 – 43)·

(5 – 23) (3 – 43)

= –39

= 39 – 263–39

–1


(5 – 23) (3 + 43)


(3)2 – (43)2 = 9 – 48 = –39

= –3915 – 63 – 203 + 24

= 13(3 – 23)–393

= –3 + 23 3

Exercise A. Simplify.1. 12 (3 + 32) 2. 8 (3 + 312)3. 6 (43 – 52) 4. 20 (45 – 5)5. (3 – 22)(2 + 32) 6. (5 – 23)(2 + 3)

15. (33 – 22)(23 + 32) 16. (22 –5)(42 + 35)

7. (3 – 5) (23+ 3) 8. (26 – 3) (26 + 3)

9. (43 – 2) (43 + 2) 10. (52 + 3) (52 – 3) 11. (23 – 5) (23 + 5) 12. (23 + 5) (23 + 5) 13. (43 – 2) (43 – 2) 14. (52 + 3) (52 + 3)

17. (25 – 23)(45 –5) 18. (27 –3)(47 + 33)

19. (4x – 2) (4x + 2) 20. (5x + 3) (5x – 3) 21. (4x + 2) (4x + 2) 22. (5x + 3) (5x + 3) 23. (x + h – x ) (x + h + x)


Exercise B. Divide. Rationalize the denominator.

24. 1 – 3 1 + 3

25. 5 + 2 3 – 2

26. 1 – 33 2 + 3

27. 1 – 53 4 + 23

28. 32 – 33 22 – 43

29. 25 + 22 34 – 32

30. 42 – 37 22 – 27

31. x + 3x – 3

32. 3x – 33x + 2

33. x – 2x + 2 + 2

34. x – 4x – 3 – 1


4 4 more on algebra of radicals-x

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