MATHS SYMBOLS - PROPERTIES of EXPONENTS - EXPONENTIATION - a SUPERSCRIPT n - 0 SUPERSCRIPT n - 1 SUPERSCRIPT n - 6 PROPERTIES - SECOND PROPERTY - FOURTH PROPERTY - PROOFS and EXAMPLES

1. ENZO EXPOSYTO
MATHS
SYMBOLS
PROPERTIES of EXPONENTS
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2. 23 22 21 20
EXPONENTS
2-1 2-2 2-3 2-4
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3.
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4. 1 - Exponents and Their Properties - 1 5
2 - Exponents and Their Properties - 2 10
3 - Exponents and Their Properties - 3 13
4 - Exponents and Their Properties - 4 15
5 - SitoGraphy 19
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5. EXPONENTS
and THEIR
PROPERTIES
(1)
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6. a superscript n
an say: a superscript n;
(a^n) a is a positive number and n is a positive integer;
The base a is raised to the power of n;
an is equal to
(we write ‘a’, n times
and
the multiplication sign ‘×’, n-1 times):
an = a × a × ... × a
for example:
23 = 2 × 2 × 2 = 8
(3 times "2" and 3-1 times "x")
a1 say: a superscript 1
(a^1) It’s equal to a: a1= a;
for example:
51 = 5
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7. a superscript n super superscript m
say: a superscript n super superscript m
a is a positive number; n and m are positive integers;
a^n^m It’s equal to a^(nm):
a^n^m = a^(nm);
for example:
2^3^2 = 2^(32) = 29 = 512
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8. 0 superscript n AND 0 superscript (-n)
0n n is a positive integer;
(0^n) it’s equal to 0:
0n = 0;
for example:
05 = 0 × 0 × 0 × 0 × 0 = 0
0-n n is a positive integer;
(0^(-n)) 0-n is impossible
and the result Does Not Exist (DNE)
for example:
0-1 = 1 = 1 is impossible (it’s impossible dividing 1 by 0)
01 0
and the result Does Not Exist (DNE)
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9. 1 superscript n AND (-1) superscript n
1n It is equal to 1:
(1^n) 1n = 1;
for example:
15 = 1 × 1 × 1 × 1 × 1 = 1
(-1)n n even;
it is equal to +1;
for example:
(-1)4 = (-1) × (-1) × (-1) × (-1) = +1
(-1)n n odd;
it is equal to -1;
for example
(-1)5 = (-1) × (-1) × (-1) × (-1) × (-1) = -1
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10. EXPONENTS
and THEIR
PROPERTIES
(2)
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11. Exponents - 6 Properties - 2A
EXPONENTS and THEIR PROPERTIES - 2A
[a, b elements of R+]
[m, n elements of Z+]
Z+ = {1, 2, 3, …}
Property Powers Exponents Powers Exponents Result
1st am · an = am + n 23 · 22 = 23 + 2 = 32
2nd am
=
an
am -n
23
=
22
23 -2 = 2
3rd (am)n = am *n (23)2 = 23 *2 = 64
4th n√am = am : n 2√24 = 24 : 2 = 4
5th an · bn = (a · b)n 22 · 32 = (2 · 3)2 = 36
6th
an
———- =
bn
(_a_)n
b
43
———- =
23
(_4_)3
=
2
8
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12. Exponents - 6 Properties - 2B
EXPONENTS and THEIR PROPERTIES - 2B
[a, b elements of R+]
[m, n elements of Z+]
Z+ = {1, 2, 3, …}
Property Exponents Powers Exponents Powers Result
1st am + n = am · an
23 + 2 = 23 · 22 = 32
2nd
am -n = am
an
23 -2 = 23
=
22
2
3rd am *n = (am)n
23 *2 = (23)2 = 64
4th am : n = n√am 24 : 2 = 2√24 = 4
5th (a · b)n = an · bn (2 · 3)2 = 22 · 32 = 36
6th
(_a_)n =
b
an
———-
bn
(_4_)3
=
2
43
———- =
23
8
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13. EXPONENTS
and THEIR
PROPERTIES
(3)
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14. Exponents - 6 Properties - Proofs/Examples
PROOFS / EXAMPLES
[a, b elements of R+]
[the exponents m and n are elements of Z+]
Z+ = {1, 2, 3, …}
1st a3 · a2 = (a · a · a) · (a · a) = a · a · a · a · a = a5 = a3 + 2
2nd a3 = (a · a · a) = a = a1 = a3 - 2
a2 (a · a)
3rd (a3)2 = (a · a · a) · (a · a · a) = a · a · a · a · a · a = a6 = a3 * 2
4th 2√a4 = 2√(a · a · a · a) = a · a = a2 = a4 : 2
5th a3 · b3 = (a · a · a) · (b · b· b) = a · b · a · b · a · b = … = (a·b)3
6th
a3 (a · a · a) a a a a
——- = ————— = —— . —— . —— = … = (—)3
b3 (b · b· b) b b b b
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15. EXPONENTS
and THEIR
PROPERTIES
(4)
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16. Exponents - 2nd property- a superscript 0
EXPONENTS and THEIR PROPERTIES - 4A
Z+ = {1, 2, 3, …}
Reference Property Notice Proof Example
2nd
Property
a0 = 1
a ≠ 0
n Z+ a0 = an - n 20 = 23 - 3
= an
an
= 23
23
= 1 = 1
1 = a0 a ≠ 0 1 = 20
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17. Exponents - 2nd property - a superscript (-n)
EXPONENTS and THEIR PROPERTIES - 4B
Z+ = {1, 2, 3, …}
Reference Property Notice Proof Example
2nd
Property
a-n = 1
an
a ≠ 0
n Z+ a-n = a0-n 2-3 = 20-3
= a0
an
= 20
23
= 1
an
= 1
23
1 = a-n
an
a ≠ 0
n Z+
1 = 2-3
23
Negative exponents are
the reciprocals of the positive exponents:
an = 1
a-n
a ≠ 0
n Z+
23 = 1
2-3
1 = an
a-n
a ≠ 0
n Z+
1 = 23
2-3
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18. Exponents - 4th property - a superscript (1/n)
EXPONENTS and THEIR PROPERTIES - 4C
Z+ = {1, 2, 3, …}
Reference Property Notice Proof Example
4th
Property
n√a = a1/n
a ≥ 0
n Z+
n√a = n√a1 3√8 = 3√81
= a1/n = 81/3
a1/n = n√a
a ≥ 0
n Z+ 81/3 = 3√8
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19. SitoGraphy
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20. http://www.gh-mathspeak.com/examples/grammar-rules/?rule=scripts
http://www.rapidtables.com/math/number/exponent.htm
http://www.mathplanet.com/education/algebra-1/exponents-and-exponential-functions/properties-of-exponents
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