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1. RADICALS

2. Properties of Surds
1 1
n 4
Definition : a a n
16 4
16 2
m 1 m m
n 3 3 3
a n
a n a 16 4 4
16 2 8
m 1
m n m 3
4
163
n
a n
a a 16 4 4
4096 8
m n m
n
a a
Jeff Bivin -- LZHS

3. Simplify the surds
For positif real numbers a and b
n n n
a.b a. b
32 16.2 16. 2 4 2
3 294 3 49.6 3 49. 6 3.7. 6 21 6
3
32 3
8.4 3
8. 3 4 3
2 4
Jeff Bivin -- LZHS

4. Simplify!!!!!
288 12 2
500 x 10 5
x x x
3
3
500 5 4
Jeff Bivin -- LZHS

5. Algebraic Operation on Surds
Addition and Subtraction
Suppose , b
a R and c R
a c b c= a b c
5 5 10 5 = 5 10 5 15 5
5 3 48 2 27 = 5 3 4 3 6 3 5 4 6 3
3 3
20 125 3 5 = 2 5 5 5 3 5 = 0
Jeff Bivin -- LZHS

6. Algebraic Operation on Surds
Multiplication
Suppose , b
a R and c, d R
a c .b d = ab. cd
5 5.10 2 = 5.10 5.2 50 10
3 2 3 2 3 6 6 2 5 2 6
2 5 5 3 2 = 2.5 2.3 5.2 = 10 6 10
Jeff Bivin -- LZHS

7. Algebraic Operation on Surds
Division
Suppose , b
a R and c, d R
a c :b d = a:b c:d
1 1
5 8 :10 2 = 5 :10 8: 2 4 .2 1
2 2
2 3.3 2 6 6 1
= =
12 6 12 6 2
2 2
5 2 5 2 5 2 3
1
3 3 3
Jeff Bivin -- LZHS

8. Simplify!!!!!
0,36 0,16 1
6 2 5 2 2 5 2 11 10
a b
a b
a b
Jeff Bivin -- LZHS

9. Simplifying Surds expression
(a b) 2 ab
and
(a b) 2 ab
2
a b a b 2 ab
2
a b a b 2 ab
a b a b 2 ab
a b 2 ab a b

10. Simplifying Surds expression
(a b) 2 ab
and
(a b) 2 ab
2
a b a b 2 ab
2
a b a b 2 ab
a b a b 2 ab
a b 2 ab a b

11. Example!
5 2 6 3 2 2 3.2 3 2
8 2 15 3 5 2 3.5 5 3
Jeff Bivin -- LZHS

12. Try this Questions!!!
3 2 2 1 2
7 2 10 5 2
8 48 6 2
14 6 5 9 5
0,3
1
0, 08
1
a b
15
....
Jeff Bivin -- LZHS
a b

13. Rationalizing Fractions Involving
Surds
Consider!!
2 2
a b a b a b
a a a Conjugate
2 2
a b a b a b a b
2
2
a b a b a b a2 b
Jeff Bivin -- LZHS

14. Simplify
6 6
3
2 2
Jeff Bivin -- LZHS

15. Simplify
10 10 3 30 30
3 3 3 9 3
Jeff Bivin -- LZHS

16. Simplify
24 24 12 12 2 3 5
10 10 5 5 5 5
2 15 2 15
25 5
Jeff Bivin -- LZHS

17. Simplify
1 3 2 3 2
3 2 3 2 9 3 2 3 2 2
3 2
7
Jeff Bivin -- LZHS

18. Simplify
3 4 5 34 5
4 5 4 5 16 4 5 4 5 5
34 5 12 3 5
11 11
Jeff Bivin -- LZHS

19. Simplify
2 1 7 21 7
1 7 1 7 1 7 7 7
21 7 1 7
6 3
Jeff Bivin -- LZHS

20. Simplify
3 2 3 2 3 3
2 3 2 3 4 2 3 2 3 3
2 3 3
2 3 3
1
Jeff Bivin -- LZHS

21. Simplify
3 2 3 2 3 2 3
1 1 2 2 2 4
3 3
2 2 3
2 2 3
2 3 2 2
Jeff Bivin -- LZHS

22. Simplify
5 5 4 5 4 5
5
2 2 3 2 3 162
5
3 3 5 4
3 5 5
3 3
Jeff Bivin -- LZHS

23. Try this Questions!!!
1 1
2
2 2 4
1 3
...
4 12
5 6
a b 30 9
5 6
a b ......?
Jeff Bivin -- LZHS

24. Exponential Equations
Properties
B C
a a B C
3x 27 3x 33 x 3
1 1 1 1
1 2x 1 2 52 x 5 2 2x x
52 x 5 5 5 5 2 4
5
1
3
36 x 3
27 x 1
Jeff Bivin -- LZHS

25. Try this Questions!!!
x 2 1
5 125 x
2
4x 2 3
16 x 5 x 4
1 5
32 2x
x
273 x 7
2
1
x 3
2
0, 09 x 2
1
0,33 x 1
Jeff Bivin -- LZHS

26. To be continued…