ROOTS - PROPERTIES 0, 1, 1A, 1B, 1C, 1D, 2, 2A - ROOTS and THEIR PROPERTIES - 10+ PROPERTIES - RATIONALISING the DENOMINATOR - RADICAL EQUATION - PROOFS, EXAMPLES - CALCULATIONS STEP by STEP MATHS SYMBOLS www.enzoexposito.it/mobile/matematica.html www.enzoexposito.it/mobile/aritmetica.html

1. ENZO EXPOSYTO
MATHS
SYMBOLS
ROOTS and THEIR PROPERTIES
Enzo Exposyto 1

2. 2√16 2√9 2√4 2√1 2√0
ROOTS
3√-27 3√-8 3√-1 3√0
Enzo Exposyto 2

3.
Enzo Exposyto 3

4. 1 - Squares and Square Roots 6
2 - Definitions 10
3 - Roots and Their Properties 20
4 - Property 0 22
5 - Property 1 25
6 - Property 2 31
7 - Property 3 34
8 - Property 4 38
9 - Property 5 42
Enzo Exposyto 4

5. 10 - Property 6 46
11 - Property 7 51
12 - Property 8 55
13 - Property 9 - Rationalising the Denominator 63
14 - Equation 76
15 - SitoGraphy 78
Enzo Exposyto 5

6. ROOTS
and
THEIR
PROPERTIES
Enzo Exposyto 20

7. REMEMBERING
the 4th Property of Exponents:
Example:
and the Property
Example:
2√4 = 4½
Enzo Exposyto 21

8. we get …
PROPERTY 0)
(n√a)m = n√am
and
(n√am)p = n√am*p
IF n IS EVEN,
a must be ≥ 0
Enzo Exposyto 22

9. PROPERTY 0)
Proof
(n√a)m = (a1/n)m
= am/n
= n√am
and
(n√am)p = (am/n)p
= am*p/n
= n√am*p
Enzo Exposyto 23

10. PROPERTY 0)
Example:
(2√4)3 = 2√43
23 = 2√64
8 = 8
and
(2√24)3 = 2√24*3
(2√16)3 = 2√24*3
43 = 2√212
64 = 212/2
64 = 26
64 = 64
Enzo Exposyto 24

11. PROPERTY 1)
n√a * n√b = n√(a*b)
IF n IS EVEN,
a and b
must both be
≥ 0
Enzo Exposyto 25

12. PROPERTY 1)
Proof
n√a * n√b = a1/n * b1/n
= (a*b)1/n
= n√(a*b)
Example:
2√4 *
2√9 = 2√(4 * 9)
2 * 3 = 2√36
6 = 6
Enzo Exposyto 26

13. PROPERTY 1A)
From PROPERTY 1)
we get
IF n IS EVEN,
a and b
must both be
≥ 0
Example:
Enzo Exposyto 27

14. PROPERTY 1B)
From PROPERTY 1)
we get
a ≥ 0
Example:
√4 * √4 = √(4 * 4) = √42 = (42)½ = 42/2 = 4
Enzo Exposyto 28

15. PROPERTY 1C)
From PROPERTY 1)
we get
Example:
3√8 *
3√8 *
3√8 = 3√(8 * 8 * 8) = 3√83 = (83)1/3 = 83/3 = 8
Enzo Exposyto 29

16. PROPERTY 1D)
From PROPERTY 1)
we get
IF n IS EVEN,
a must be
≥ 0
Example:
5√3 *
5√3 *
5√3 *
5√3 *
5√3 = ... = 5√35 = (35)1/5 = 35/5 = 3
Enzo Exposyto 30

17. PROPERTY 2)
IF n IS EVEN,
a must be ≥ 0
and
b must be > 0
IF n IS ODD,
a can be < = > 0
and
b must be < > 0
b CAN’T BE ZERO,
since we can't divide by zero
Enzo Exposyto 31

18. PROPERTY 2)
PROOF
n√( a ) = ( a )1/n
b b
= a1/n
b1/n
= n√a
n√b
Example:
Enzo Exposyto 32

19. PROPERTY 2A)
From PROPERTY 2)
we get
n√a = n√( a )
n√b b
IF n IS EVEN,
a must be ≥ 0
and
b must be > 0
IF n IS ODD,
a can be < = > 0
and
b must be < > 0
b CAN’T BE ZERO,
since we can't divide by zero
Enzo Exposyto 33