69
Algebra of Radicals
Algebra of Radicals
Algebra of RadicalsMultiplication Rule: x·y = x·y, x·x = x
Algebra of RadicalsMultiplication Rule: x·y = x·y, x·x = x
Division Rule: yx
yx =
Algebra of RadicalsMultiplication Rule: x·y = x·y, x·x = x
Division Rule: yx
yx =
Example A. Simplify.a. 3 * 3
Algebra of RadicalsMultiplication Rule: x·y = x·y, x·x = x
Division Rule: yx
yx =
Example A. Simplify.a. 3 * 3 = 3
Algebra of RadicalsMultiplication Rule: x·y = x·y, x·x = x
Division Rule: yx
yx =
Example A. Simplify.a. 3 * 3 = 3b. 33 * 3
3
Algebra of RadicalsMultiplication Rule: x·y = x·y, x·x = x
Division Rule: yx
yx =
Example A. Simplify.a. 3 * 3 = 3b. 33 * 3 = 3 * 3
3
Algebra of RadicalsMultiplication Rule: x·y = x·y, x·x = x
Division Rule: yx
yx =
Example A. Simplify.a. 3 * 3 = 3b. 33 * 3 = 3 * 3 = 9
c. (3*3)23
Algebra of RadicalsMultiplication Rule: x·y = x·y, x·x = x
Division Rule: yx
yx =
Example A. Simplify.a. 3 * 3 = 3b. 33 * 3 = 3 * 3 = 9
c. (3*3)2 = 3*3 * 3 *33
Algebra of RadicalsMultiplication Rule: x·y = x·y, x·x = x
Division Rule: yx
yx =
Example A. Simplify.a. 3 * 3 = 3b. 33 * 3 = 3 * 3 = 9
c. (3*3)2 = 3*3 * 3 *33
3
Algebra of RadicalsMultiplication Rule: x·y = x·y, x·x = x
Division Rule: yx
yx =
Example A. Simplify.a. 3 * 3 = 3b. 33 * 3 = 3 * 3 = 9
c. (3*3)2 = 3*3 * 3 *3 = 3 * 3 * 33
3
Algebra of RadicalsMultiplication Rule: x·y = x·y, x·x = x
Division Rule: yx
yx =
Example A. Simplify.a. 3 * 3 = 3b. 33 * 3 = 3 * 3 = 9
c. (3*3)2 = 3*3 * 3 *3 = 3 * 3 * 3 = 273
3
Algebra of RadicalsMultiplication Rule: x·y = x·y, x·x = x
Division Rule: yx
yx =
Example A. Simplify.a. 3 * 3 = 3b. 33 * 3 = 3 * 3 = 9
c. (3*3)2 = 3*3 * 3 *3 = 3 * 3 * 3 = 273
3
Algebra of RadicalsMultiplication Rule: x·y = x·y, x·x = x
Division Rule: yx
yx =
Example A. Simplify.a. 3 * 3 = 3b. 33 * 3 = 3 * 3 = 9
d. 12 * 3
c. (3*3)2 = 3*3 * 3 *3 = 3 * 3 * 3 = 273
3
Algebra of RadicalsMultiplication Rule: x·y = x·y, x·x = x
Division Rule: yx
yx =
Example A. Simplify.a. 3 * 3 = 3b. 33 * 3 = 3 * 3 = 9
d. 12 * 3 = 36
c. (3*3)2 = 3*3 * 3 *3 = 3 * 3 * 3 = 273
3
Algebra of RadicalsMultiplication Rule: x·y = x·y, x·x = x
Division Rule: yx
yx =
Example A. Simplify.a. 3 * 3 = 3b. 33 * 3 = 3 * 3 = 9
d. 12 * 3 = 36 = 6
c. (3*3)2 = 3*3 * 3 *3 = 3 * 3 * 3 = 273
3
Algebra of RadicalsMultiplication Rule: x·y = x·y, x·x = x
Division Rule: yx
yx =
Example A. Simplify.a. 3 * 3 = 3b. 33 * 3 = 3 * 3 = 9
d. 12 * 3 = 36 = 6e. 108 *75
c. (3*3)2 = 3*3 * 3 *3 = 3 * 3 * 3 = 273
3
Algebra of RadicalsMultiplication Rule: x·y = x·y, x·x = x
Division Rule: yx
yx =
Example A. Simplify.a. 3 * 3 = 3b. 33 * 3 = 3 * 3 = 9
d. 12 * 3 = 36 = 6e. 108 *75 = 36*3 * 25*3
c. (3*3)2 = 3*3 * 3 *3 = 3 * 3 * 3 = 273
3
Algebra of RadicalsMultiplication Rule: x·y = x·y, x·x = x
Division Rule: yx
yx =
Example A. Simplify.a. 3 * 3 = 3b. 33 * 3 = 3 * 3 = 9
d. 12 * 3 = 36 = 6e. 108 *75 = 36*3 * 25*3 = 63 * 53
c. (3*3)2 = 3*3 * 3 *3 = 3 * 3 * 3 = 273
3
Algebra of RadicalsMultiplication Rule: x·y = x·y, x·x = x
Division Rule: yx
yx =
Example A. Simplify.a. 3 * 3 = 3b. 33 * 3 = 3 * 3 = 9
d. 12 * 3 = 36 = 6e. 108 *75 = 36*3 * 25*3 = 63 * 53
3
c. (3*3)2 = 3*3 * 3 *3 = 3 * 3 * 3 = 273
3
Algebra of RadicalsMultiplication Rule: x·y = x·y, x·x = x
Division Rule: yx
yx =
Example A. Simplify.a. 3 * 3 = 3b. 33 * 3 = 3 * 3 = 9
d. 12 * 3 = 36 = 6e. 108 *75 = 36*3 * 25*3 = 63 * 53 = 30*3
3
c. (3*3)2 = 3*3 * 3 *3 = 3 * 3 * 3 = 273
3
Algebra of RadicalsMultiplication Rule: x·y = x·y, x·x = x
Division Rule: yx
yx =
Example A. Simplify.a. 3 * 3 = 3b. 33 * 3 = 3 * 3 = 9
d. 12 * 3 = 36 = 6e. 108 *75 = 36*3 * 25*3 = 63 * 53 = 30*3 = 90
3
c. (3*3)2 = 3*3 * 3 *3 = 3 * 3 * 3 = 273
3
Algebra of RadicalsMultiplication Rule: x·y = x·y, x·x = x
Division Rule: yx
yx =
Example A. Simplify.a. 3 * 3 = 3b. 33 * 3 = 3 * 3 = 9
d. 12 * 3 = 36 = 6e. 108 *75 = 36*3 * 25*3 = 63 * 53 = 30*3 = 90
3f. 33 2 * 2 *2 3 2
c. (3*3)2 = 3*3 * 3 *3 = 3 * 3 * 3 = 273
3
Algebra of RadicalsMultiplication Rule: x·y = x·y, x·x = x
Division Rule: yx
yx =
Example A. Simplify.a. 3 * 3 = 3b. 33 * 3 = 3 * 3 = 9
d. 12 * 3 = 36 = 6e. 108 *75 = 36*3 * 25*3 = 63 * 53 = 30*3 = 90
3f. 33 2 * 2 *2 3 2 = 3 * 2 * 3 3 2 2 2
c. (3*3)2 = 3*3 * 3 *3 = 3 * 3 * 3 = 273
3
Algebra of RadicalsMultiplication Rule: x·y = x·y, x·x = x
Division Rule: yx
yx =
Example A. Simplify.a. 3 * 3 = 3b. 33 * 3 = 3 * 3 = 9
d. 12 * 3 = 36 = 6e. 108 *75 = 36*3 * 25*3 = 63 * 53 = 30*3 = 90
3f. 33 2 * 2 *2 3 2 = 3 * 2 * 3 3 2 2 2
3 2
c. (3*3)2 = 3*3 * 3 *3 = 3 * 3 * 3 = 273
3
Algebra of RadicalsMultiplication Rule: x·y = x·y, x·x = x
Division Rule: yx
yx =
Example A. Simplify.a. 3 * 3 = 3b. 33 * 3 = 3 * 3 = 9
d. 12 * 3 = 36 = 6e. 108 *75 = 36*3 * 25*3 = 63 * 53 = 30*3 = 90
3f. 33 2 * 2 *2 3 2 = 3 * 2 * 3 3 2 2 2
3 2
= 3 * 3 * 2 * 2 2
c. (3*3)2 = 3*3 * 3 *3 = 3 * 3 * 3 = 273
3
Algebra of RadicalsMultiplication Rule: x·y = x·y, x·x = x
Division Rule: yx
yx =
Example A. Simplify.a. 3 * 3 = 3b. 33 * 3 = 3 * 3 = 9
d. 12 * 3 = 36 = 6e. 108 *75 = 36*3 * 25*3 = 63 * 53 = 30*3 = 90
3f. 33 2 * 2 *2 3 2 = 3 * 2 * 3 3 2 2 2
3 2
= 3 * 3 * 2 * 2 2 = 362
Example B. Simplify.
18x
x3
2
1415
35 3
Algebra of Radicals
a. =
b. =
To simplify a radical expression we may simplify the radicands first before we extract any root.
Example B. Simplify.
18x
x3
2 18
xx32
1415
35 3
Algebra of Radicals
a. =b.
=
To simplify a radical expression we may simplify the radicands first before we extract any root.
Example B. Simplify.
18x
x3
2 18 9
xx3 x22
1415
35 3
Algebra of Radicals
a. =b.
=
To simplify a radical expression we may simplify the radicands first before we extract any root.
Example B. Simplify.
18x
x3
2 18 9
xx3 x22
9x21
3x1
1415
35 3
Algebra of Radicals
a. = = =
b. =
To simplify a radical expression we may simplify the radicands first before we extract any root.
Example B. Simplify.
18x
x3
2 18 9
xx3 x22
9x21
3x1
1415
35 3
1415
35 3
Algebra of Radicals
a. = = =
b. =
To simplify a radical expression we may simplify the radicands first before we extract any root.
Example B. Simplify.
18x
x3
2 18 9
xx3 x22
9x21
3x1
1415
35 3
1415
35 3
2*7*73*3
Algebra of Radicals
a. = = =
b. ==
To simplify a radical expression we may simplify the radicands first before we extract any root.
Example B. Simplify.
18x
x3
2 18 9
xx3 x22
9x21
3x1
1415
35 3
1415
35 3
2*7*73*3
Algebra of Radicals
a. = = =
b. == 7
3= 2*21*2
To simplify a radical expression we may simplify the radicands first before we extract any root.
Example B. Simplify.
18x
x3
2 18 9
xx3 x22
9x21
3x1
1415
35 3
1415
35 3
2*7*73*3
1432
Algebra of Radicals
a. = = =
b. == 7
3= 2*21*2
= 7*232 =
To simplify a radical expression we may simplify the radicands first before we extract any root.
Example B. Simplify.
18x
x3
2 18 9
xx3 x22
9x21
3x1
1415
35 3
1415
35 3
2*7*73*3
1432
Algebra of Radicals
a. = = =
b. == 7
3= 2*21*2
= 7*232 =
We simplify higher roots in similar manner, i.e. extract as much out of the radical as possible.
To simplify a radical expression we may simplify the radicands first before we extract any root.
Algebra of RadicalsExample C. Simplify.a. 543
Algebra of RadicalsExample C. Simplify.a. 54 = √27 * 23 3
Algebra of RadicalsExample C. Simplify.a. 54 = √27 * 2 = 3√23 3 3
Algebra of RadicalsExample C. Simplify.a. 54 = √27 * 2 = 3√23 3 3
b. x3y53
Algebra of RadicalsExample C. Simplify.a. 54 = √27 * 2 = 3√23 3 3
b. x3y5 = √x3xy3y23 3
Algebra of RadicalsExample C. Simplify.a. 54 = √27 * 2 = 3√23 3 3
b. x3y5 = √x3xy3y2 = xy√xy23 33
Algebra of RadicalsExample C. Simplify.a. 54 = √27 * 2 = 3√23 3 3
c. 16x4y73
b. x3y5 = √x3xy3y2 = xy√xy23 33
Algebra of RadicalsExample C. Simplify.a. 54 = √27 * 2 = 3√23 3 3
c. 16x4y7 = √8*2x3xy6y3 3
b. x3y5 = √x3xy3y2 = xy√xy23 33
Algebra of RadicalsExample C. Simplify.a. 54 = √27 * 2 = 3√23 3 3
c. 16x4y7 = √8*2x3xy6y = 2xy2√2xy3 33
b. x3y5 = √x3xy3y2 = xy√xy23 33
Identical radicals may be added and subtracted.
Algebra of RadicalsExample C. Simplify.a. 54 = √27 * 2 = 3√23 3 3
c. 16x4y7 = √8*2x3xy6y = 2xy2√2xy3 33
b. x3y5 = √x3xy3y2 = xy√xy23 33
Identical radicals may be added and subtracted. Different radicals may not be combined.
Algebra of RadicalsExample C. Simplify.a. 54 = √27 * 2 = 3√23 3 3
c. 16x4y7 = √8*2x3xy6y = 2xy2√2xy3 33
b. x3y5 = √x3xy3y2 = xy√xy23 33
Identical radicals may be added and subtracted. Different radicals may not be combined. Example D. Simplify.a. 3 – 33 = –2√3
Algebra of RadicalsExample C. Simplify.a. 54 = √27 * 2 = 3√23 3 3
c. 16x4y7 = √8*2x3xy6y = 2xy2√2xy3 33
b. x3y5 = √x3xy3y2 = xy√xy23 33
Identical radicals may be added and subtracted. Different radicals may not be combined. Example D. Simplify.a. 3 – 33 = –2√3 b. 3 – 4√2 + 7√3 – 5√2
Algebra of RadicalsExample C. Simplify.a. 54 = √27 * 2 = 3√23 3 3
c. 16x4y7 = √8*2x3xy6y = 2xy2√2xy3 33
b. x3y5 = √x3xy3y2 = xy√xy23 33
Identical radicals may be added and subtracted. Different radicals may not be combined. Example D. Simplify.a. 3 – 33 = –2√3 b. 3 – 4√2 + 7√3 – 5√2 = 8√3 – 9√2
Algebra of RadicalsExample C. Simplify.a. 54 = √27 * 2 = 3√23 3 3
c. 16x4y7 = √8*2x3xy6y = 2xy2√2xy3 33
b. x3y5 = √x3xy3y2 = xy√xy23 33
Identical radicals may be added and subtracted. Different radicals may not be combined.
d. 2 + 5√6
Example D. Simplify.a. 3 – 33 = –2√3 b. 3 – 4√2 + 7√3 – 5√2 = 8√3 – 9√2
Algebra of RadicalsExample C. Simplify.a. 54 = √27 * 2 = 3√23 3 3
c. 16x4y7 = √8*2x3xy6y = 2xy2√2xy3 33
b. x3y5 = √x3xy3y2 = xy√xy23 33
Identical radicals may be added and subtracted. Different radicals may not be combined.
d. 2 + 5√6 = 2 + 5√6 , they can’t be combined because they are not like-terms.
Example D. Simplify.a. 3 – 33 = –2√3 b. 3 – 4√2 + 7√3 – 5√2 = 8√3 – 9√2
Algebra of RadicalsExample C. Simplify.a. 54 = √27 * 2 = 3√23 3 3
c. 16x4y7 = √8*2x3xy6y = 2xy2√2xy3 33
b. x3y5 = √x3xy3y2 = xy√xy23 33
Identical radicals may be added and subtracted. Different radicals may not be combined.
d. 2 + 5√6 = 2 + 5√6 , they can’t be combined because they are not like-terms.
Example D. Simplify.a. 3 – 33 = –2√3 b. 3 – 4√2 + 7√3 – 5√2 = 8√3 – 9√2
e. 2√12 – 5√27 + √18
Algebra of RadicalsExample C. Simplify.a. 54 = √27 * 2 = 3√23 3 3
c. 16x4y7 = √8*2x3xy6y = 2xy2√2xy3 33
b. x3y5 = √x3xy3y2 = xy√xy23 33
Identical radicals may be added and subtracted. Different radicals may not be combined.
d. 2 + 5√6 = 2 + 5√6 , they can’t be combined because they are not like-terms.
Example D. Simplify.a. 3 – 33 = –2√3 b. 3 – 4√2 + 7√3 – 5√2 = 8√3 – 9√2
e. 2√12 – 5√27 + √18 = 2√4*3 – 5√9*3 + √9*2
Algebra of RadicalsExample C. Simplify.a. 54 = √27 * 2 = 3√23 3 3
c. 16x4y7 = √8*2x3xy6y = 2xy2√2xy3 33
b. x3y5 = √x3xy3y2 = xy√xy23 33
Identical radicals may be added and subtracted. Different radicals may not be combined.
d. 2 + 5√6 = 2 + 5√6 , they can’t be combined because they are not like-terms.
Example D. Simplify.a. 3 – 33 = –2√3 b. 3 – 4√2 + 7√3 – 5√2 = 8√3 – 9√2
e. 2√12 – 5√27 + √18 = 2√4*3 – 5√9*3 + √9*2= 2*2√3 – 5*3√3 + 3√2
Algebra of RadicalsExample C. Simplify.a. 54 = √27 * 2 = 3√23 3 3
c. 16x4y7 = √8*2x3xy6y = 2xy2√2xy3 33
b. x3y5 = √x3xy3y2 = xy√xy23 33
Identical radicals may be added and subtracted. Different radicals may not be combined.
d. 2 + 5√6 = 2 + 5√6 , they can’t be combined because they are not like-terms.
Example D. Simplify.a. 3 – 33 = –2√3 b. 3 – 4√2 + 7√3 – 5√2 = 8√3 – 9√2
e. 2√12 – 5√27 + √18 = 2√4*3 – 5√9*3 + √9*2= 2*2√3 – 5*3√3 + 3√2 = 4√3 – 15√3 + 3√2
Algebra of RadicalsExample C. Simplify.a. 54 = √27 * 2 = 3√23 3 3
c. 16x4y7 = √8*2x3xy6y = 2xy2√2xy3 33
b. x3y5 = √x3xy3y2 = xy√xy23 33
Identical radicals may be added and subtracted. Different radicals may not be combined.
d. 2 + 5√6 = 2 + 5√6 , they can’t be combined because they are not like-terms.
Example D. Simplify.a. 3 – 33 = –2√3 b. 3 – 4√2 + 7√3 – 5√2 = 8√3 – 9√2
e. 2√12 – 5√27 + √18 = 2√4*3 – 5√9*3 + √9*2= 2*2√3 – 5*3√3 + 3√2 = 4√3 – 15√3 + 3√2 = –11√3 + 3√2
Algebra of Radicals
e. 23 – 24
Algebra of Radicals
simplify each radicale. 23 – 24
Algebra of Radicals
e. 23 – 24 simplify each radical
2*23*2 – 4*6=
extract denominator extract perfect–square
Algebra of Radicals
e. 23 – 24 simplify each radical
2*23*2 – 4*6=
2*26 – 26=
Algebra of Radicals
2 simplify each radical
2*23*2 – 4*6=
2*26 – 26=
21= 6 – 26
e. 23 – 24
Algebra of Radicals
2 simplify each radical
2*23*2 – 4*6=
2*26 – 26=
21= 6 – 26
2–3= 6 2
–36or
e. 23 – 24
Algebra of Radicals
2 simplify each radical
2*23*2 – 4*6=
2*26 – 26=
21= 6 – 26
2–3= 6 2
–36or
e. 23 – 24
Extracting the denominator out of the radical sign enable us to identify who can be combined with whom.
Algebra of Radicals
2 simplify each radical
2*23*2 – 4*6=
2*26 – 26=
21= 6 – 26
2–3= 6 2
–36or
e. 23 – 24
Extracting the denominator out of the radical sign enable us to identify who can be combined with whom. For instance
21 = 0.5
Algebra of Radicals
2 simplify each radical
2*23*2 – 4*6=
2*26 – 26=
21= 6 – 26
2–3= 6 2
–36or
e. 23 – 24
Extracting the denominator out of the radical sign enable us to identify who can be combined with whom. For instance
21 = 0.5 is really (half of root 2),2
1 2
Algebra of Radicals
2 simplify each radical
2*23*2 – 4*6=
2*26 – 26=
21= 6 – 26
2–3= 6 2
–36or
e. 23 – 24
Extracting the denominator out of the radical sign enable us to identify who can be combined with whom. For instance
21 = 0.5 is really (half of root 2), hence2
1 2
2 –
= 21 2 .22
1
Algebra of Radicals
2 simplify each radical
2*23*2 – 4*6=
2*26 – 26=
21= 6 – 26
2–3= 6 2
–36or
e. 23 – 24
Extracting the denominator out of the radical sign enable us to identify who can be combined with whom. For instance
21 = 0.5 is really (half of root 2), hence2
1 2
2 –
= 21 2 .22
1simplified when there is still any radical in denominator.
Therefore a fraction is considered not
Algebra of RadicalsExercise A. Simplify.
7. (2*2 )2
1. 32 * 2 2. 33 * 2 3. 33 * 2 *3
8. (2*3 )2 9. (3*3 )2
10. (–2*4 )2 11. (–4*2 )2 12. (–3*5 )2
4. 2(2 )2 5. 2(3 )2 6. 3(–3 )2
13. (–2*3 ) (3*3 ) 14. (4*2 ) 2(3 )
15. (–5*5 ) (5 ) 16. (4*2 ) (22 )
17. 22 – (3)2
19. (4*3)2 – 72
18. (2)2 + (3)2
20. (3*3 )2 – (–3 )2
21. (–5*5 )2 – (5 )2 22. (–5*5 )2 – (–5 )2
Algebra of RadicalsExercise B. Simplify each radical then combine.23. 12 – 33
Example C. Simplify.
35. 163
42. 16x5y11341. x6y43
36. – 243 37. –323
38. 813 39. –1083 40. 1283
43. 81x9y83
24. 48 – 518 25. 427 – 512 26. 48 – 518 + 750
27. 248 – 375 + 232 – 8
28. (32)(23) – (53)(42)
29. (32)(22) – (23)(32) + (22)(23) – (23)(43)
30. (43)(33) – (43)(25) + (55)(33) – (25)(45)
31. 1/8 –2 32. 21/27 – 23 33. 62/3 – 26
