ROOTS - PROPERTIES 8, 8A, 8B, 8C - 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 8)
a * n√b = n√(an * b)
IF n IS EVEN,
a and b
must both be
≥ 0
Enzo Exposyto 55

8. PROPERTY 8)
Proof
a * n√b = an/n * b1/n
= an*1/n * b1/n
= (an)1/n * b1/n
= (an * b)1/n
= n√(an * b)
Enzo Exposyto 56

9. PROPERTY 8)
Example:
3 * 2√4 = 2√(32 * 4)
3 * 2 = 2√(9 * 4)
6 = 2√36
6 = 6
Enzo Exposyto 57

10. PROPERTY 8A)
From PROPERTY 8)
we get
n√(an * b) = a * n√b
IF n IS EVEN,
a and b
must both be
≥ 0
Enzo Exposyto 58

11. PROPERTY 8B)
Iff
(If and Only If)
m = n+p,
then
n√am = n√an+p = a n√ap
IF n IS EVEN,
a must be
≥ 0
Enzo Exposyto 59

12. PROPERTY 8B)
Proof
m = n+p
n√am = n√an+p
= n√(an * ap)
= (an * ap)1/n
= (an)1/n * (ap)1/n
= an*1/n * ap*1/n
= an/n * ap/n
= a * ap/n
= a n√ap
Enzo Exposyto 60

13. PROPERTY 8B)
Example:
2√26 = 2√22+4 = 2 2√24
2√26 = 2 2√24
2√64 = 2 2√24
8 = 2 2√16
8 = 2 * 4
8 = 8
Enzo Exposyto 61

14. PROPERTY 8C)
From PROPERTY 8B)
we get
(m = n+p)
a n√ap = n√an+p = n√am
IF n IS EVEN,
a must be
≥ 0
Enzo Exposyto 62