ROOTS - PROPERTIES 3, 3A, 4, 4A, 5, 5A - 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. PROPERTY 3)
n√a * m√b = n*m√am * m*n√bn = m*n√(am * bn)
IF n OR m IS EVEN,
a and b
must both be
≥ 0
Enzo Exposyto 34

8. PROPERTY 3)
Proof
n√a * m√b = a1/n * b1/m
= (a1/n)m/m
* (b1/m)n/n
= am/(m
*
n)
* bn/(m
*
n)
= (am)1/(m
*
n)
* (bn)1/(m
*
n)
= (am
* bn)1/(m
*
n)
= m*n√(am
* bn)
Enzo Exposyto 35

9. PROPERTY 3)
Example:
2√4 * 3√8 = 3*2√(43 * 82)
2 * 2 = 6√(64 * 64)
4 = 6√642
4 = 3√64
4 = 4
Enzo Exposyto 36

10. PROPERTY 3A)
From PROPERTY 3)
we get
m
*
n√(am
* bn) = m
*
n√am
*
m
*
n√bn = n√a *
m√b
IF n OR m IS EVEN,
a and b
must both be
≥ 0
Enzo Exposyto 37

11. PROPERTY 4)
n√a = m*n√am = m*n√(am)
m√b m*n√bn bn
b must be < > 0
IF n OR m IS EVEN,
a must be ≥ 0
and
b > 0
Enzo Exposyto 38

12. PROPERTY 4)
Proof
n√a = a1/n
m√b b1/m
= a(1/n)m/m
b(1/m)n/n
= am/(m*n)
bn/(m*n)
= (am)1/(m*n)
(bn)1/(m*n)
= (am)1/(m*n)
bn
= m*n√(am)
bn
Enzo Exposyto 39

13. PROPERTY 4)
Example:
2√4 = 3*2√(43)
3√8 82
2 = 6√(43)
2 82
2 = 6√(64)
2 64
1 = 6√1
1 = 1
Enzo Exposyto 40

14. PROPERTY 4A)
From PROPERTY 4)
we get
m*n√(am) = m*n√am = n√a
bn m*n√bn m√b
b must be < > 0
IF n OR m IS EVEN,
a must be ≥ 0
and
b > 0
Enzo Exposyto 41

15. PROPERTY 5)
n√a * m√a = m*n√am+n
IF n OR m IS EVEN,
a must be
≥ 0
Enzo Exposyto 42

16. PROPERTY 5)
Proof
n√a * m√a = a1/n * a1/m
= a1/n + 1/m
= a(m+n)/(m*n)
= m*n√am+n
Enzo Exposyto 43

17. PROPERTY 5)
Example:
2√16 * 4√16 = 4*2√164+2
4 * 2 = 8√166
8 = 8√(24)6
8 = 8√224
8 = 224/8
8 = 23
8 = 8
Enzo Exposyto 44

18. PROPERTY 5A)
From PROPERTY 5)
we get
m*n√am+n = n√a * m√a
IF n OR m IS EVEN,
a must be
≥ 0
Enzo Exposyto 45