Petita secció amb fórmules definicions i propietats purament.
Si teniu curiositat per veure una introducció rigorosa amb demostracions, podeu consultar les demostracions de cada una de les fórmules, en la direcció de wikipedia anomenada potenciació .
a
⋅
1
b
=
1
b
⋅
a
=
a
b
{\displaystyle a\cdot {\frac {1}{b}}={\frac {1}{b}}\cdot a={\frac {a}{b}}}
a
b
⋅
c
d
=
a
⋅
c
b
⋅
d
{\displaystyle {\frac {a}{b}}\cdot {\frac {c}{d}}={\frac {a\cdot c}{b\cdot d}}}
a
⋅
c
c
⋅
d
=
a
⋅
c
c
⋅
d
=
a
d
{\displaystyle {\frac {a\cdot c}{c\cdot d}}={\frac {a\cdot \color {red}{\cancel {c}}}{\color {red}{\cancel {c}}\color {black}\cdot d}}={\frac {a}{d}}}
a
b
÷
c
d
=
a
b
⋅
d
c
{\displaystyle {\frac {a}{b}}\div {\frac {c}{d}}={\frac {a}{b}}\cdot {\frac {d}{c}}}
Definició de potència amb nombres naturals com a exponent:
a
1
=
a
a
2
=
a
⋅
a
⋮
a
n
=
a
⋅
.
.
.
⋅
a
⋮
{\displaystyle {\begin{array}{|c l|}\hline a^{1}=&a\\a^{2}=&a\cdot a\\\vdots &\\a^{n}=&a\,\cdot \,...\,\cdot \,a\\\vdots &\\\hline \end{array}}}
Propietats
a
n
⋅
a
m
=
a
n
+
m
{\displaystyle a^{n}\cdot a^{m}=a^{n+m}}
(
a
n
)
m
=
a
n
⋅
m
{\displaystyle \left(a^{n}\right)^{m}=a^{n\cdot m}}
(
a
⋅
b
)
n
=
a
n
⋅
b
n
{\displaystyle \left(a\cdot b\right)^{n}=a^{n}\cdot b^{n}}
(
−
a
)
n
=
{
a
n
si n parell
−
a
n
si n imparell
{\displaystyle (-a)^{n}={\begin{cases}a^{n}&{\text{ si n parell }}\\-a^{n}&{\text{ si n imparell }}\end{cases}}}
Notació per a exponents amb valors enters:
a
−
1
=
1
a
{\displaystyle {\begin{array}{|c l|}\hline a^{-1}=&{\frac {1}{a}}\\\hline \end{array}}}
Propietats
a
−
n
=
1
a
n
{\displaystyle a^{-n}={\frac {1}{a^{n}}}}
a
n
=
1
a
−
n
{\displaystyle a^{n}={\frac {1}{a^{-n}}}}
(
a
b
)
n
=
a
n
b
n
{\displaystyle \left({\frac {a}{b}}\right)^{n}={\frac {a^{n}}{b^{n}}}}
a
n
a
m
=
a
n
−
m
=
1
a
m
−
n
{\displaystyle {\frac {a^{n}}{a^{m}}}=a^{n-m}={\frac {1}{a^{m-n}}}}
a
0
=
1
només si
a
≠
0
{\displaystyle a^{0}=1\,\,{\text{només si}}\,\,a\neq 0}
Exemples
1) Calcula el resultat de les potències següents:
a)
1
27
483
=
{\displaystyle 1^{27\;483}=}
1
⋅
1
⋅
1
⋅
.
.
.
⋅
1
⏟
1
a
p
a
r
e
i
x
24
483
v
e
g
a
d
e
s
{\displaystyle \underbrace {1\cdot 1\cdot 1\cdot \;...\;\cdot 1} _{1\;apareix\;24\;483\;vegades}}
=
1
{\displaystyle =1}
b)
0
1
000
=
{\displaystyle 0^{1\;000}=}
0
⋅
0
⋅
0
⋅
.
.
.
⋅
0
⏟
0
a
p
a
r
e
i
x
24
483
v
e
g
a
d
e
s
{\displaystyle \underbrace {0\cdot 0\cdot 0\cdot \;...\;\cdot 0} _{0\;apareix\;24\;483\;vegades}}
=
0
{\displaystyle =0}
c)
31
401
1
=
{\displaystyle 31\;401^{1}=}
31
401
{\displaystyle 31\;401}
d)
73
000
000
0
=
{\displaystyle 73\;000\;000^{0}=}
1
,
{\displaystyle 1,}
a
n
a
n
=
1
{\displaystyle {\frac {a^{n}}{a^{n}}}=1}
a
n
{\displaystyle a^{n}}
1
=
a
n
a
n
=
a
n
−
n
=
a
0
⇒
{\displaystyle 1={\frac {a^{n}}{a^{n}}}=a^{n-n}=a^{0}\Rightarrow }
1
=
a
0
{\displaystyle 1=a^{0}}
e)
(
−
1
)
30
=
{\displaystyle (-1)^{30}=}
1, perquè 30 és parell i, per tant, el signe menys s'anul·la i queda
1
31
{\displaystyle 1^{31}}
f)
(
−
1
)
31
=
{\displaystyle (-1)^{31}=}
-1, perquè 30 és imparell i, per tant, el signe menys sobreviu i queda
−
(
1
31
)
=
−
1.
{\displaystyle -(1^{31})=-1.}
g)
−
1
30
=
{\displaystyle -1^{30}=}
-1, en aquest cas l'exponent no pot afectar el signe perquè està fora, equival a escriure
−
(
1
30
)
=
−
1.
{\displaystyle -\left(1^{30}\right)=-1.}
h)
−
1
31
=
{\displaystyle -1^{31}=}
-1, en aquest cas l'exponent no pot afectar el signe perquè està fora, equival a escriure
−
(
1
31
)
=
−
1.
{\displaystyle -\left(1^{31}\right)=-1.}
1) Expressa com a una sola potència
a)
3
7
⋅
5
7
=
{\displaystyle 3^{7}\cdot 5^{7}=}
(
3
⋅
5
)
7
.
{\displaystyle (3\cdot 5)^{7}.}
b)
2
15
⋅
3
15
⋅
7
15
=
{\displaystyle 2^{15}\cdot 3^{15}\cdot 7^{15}=}
(
2
⋅
3
⋅
7
)
15
.
{\displaystyle (2\cdot 3\cdot 7)^{15}.}
c)
320
10
⋅
2
4
⋅
5
54
=
{\displaystyle 320^{10}\cdot 2^{4}\cdot 5^{54}=}
(
2
6
⋅
5
)
10
⋅
2
4
⋅
5
54
=
{\displaystyle (2^{6}\cdot 5)^{10}\cdot 2^{4}\cdot 5^{54}=}
(
2
6
)
10
⋅
5
10
⋅
2
4
⋅
5
54
=
{\displaystyle (2^{6})^{10}\cdot 5^{10}\cdot 2^{4}\cdot 5^{54}=}
2
60
⋅
5
64
⋅
2
4
=
{\displaystyle 2^{60}\cdot 5^{64}\cdot 2^{4}=}
2
64
⋅
5
64
=
{\displaystyle 2^{64}\cdot 5^{64}=}
(
2
⋅
5
)
64
=
{\displaystyle (2\cdot 5)^{64}=}
10
64
.
{\displaystyle 10^{64}.}
d)
3
7
⋅
3
8
⋅
3
0
⋅
5
0
=
{\displaystyle 3^{7}\cdot 3^{8}\cdot 3^{0}\cdot 5^{0}=}
3
7
+
8
+
0
+
0
=
{\displaystyle 3^{7+8+0+0}=}
3
15
.
{\displaystyle 3^{15}.}
2) Expressa com a producte i divisió de potències positives de nombres primers:
a)
(
28
⋅
5
)
10
=
{\displaystyle (28\cdot 5)^{10}=}
b)
(
8
⋅
10
)
8
=
{\displaystyle (8\cdot 10)^{8}=}
c)
(
50
5
2
)
7
=
{\displaystyle \left({\frac {50}{5^{2}}}\right)^{7}=}
d)
(
2
−
3
⋅
3
−
2
⋅
5
2
2
8
⋅
3
−
10
⋅
5
−
3
)
3
=
{\displaystyle \left({\frac {2^{-3}\cdot 3^{-2}\cdot 5^{2}}{2^{8}\cdot 3^{-10}\cdot 5^{-3}}}\right)^{3}=}
1)
(
2
⋅
3
)
2
=
{\displaystyle (2\cdot 3)^{2}=}
2)
(
15
3
)
3
=
{\displaystyle \left({\frac {15}{3}}\right)^{3}=}
3*)
1
2
2
⋅
2
5
=
{\displaystyle {\frac {1}{2^{2}}}\cdot 2^{5}=}
4)
(
2
⋅
5
5
)
10
=
{\displaystyle \left({\frac {2\cdot 5}{5}}\right)^{10}=}
5)
1
5
−
3
=
{\displaystyle {\frac {1}{5^{-3}}}=}
6)
5
20
5
10
=
{\displaystyle {\frac {5^{20}}{5^{10}}}=}
