Aquest és un tema suficientment conegut com per fer una exposició oberta de tots els tipus d'arrels, orientat com a recordatori, però que servirà per reforçar els possibles oblits respecte aquest contingut.
Arrels quadrades
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Presentació:
a
{\displaystyle {\sqrt {a}}}
arrel quadrada de a .
Definició: L'arrel quadrada de a és un valor real positiu r que compleix que
r
2
=
a
.
{\displaystyle r^{2}=a.}
Observació:
R
+
→
∙
R
+
0
0
=
0
1
1
=
1
4
4
=
2
100
100
=
10
0
,
01
0
,
01
=
0
,
1
250
000
250
000
=
500
{\displaystyle {\begin{matrix}\mathbb {R} ^{+}&\xrightarrow {\sqrt {\,\,\bullet \,\,}} &\mathbb {R} ^{+}\\0&&{\sqrt {0}}=&0\\1&&{\sqrt {1}}=&1\\4&&{\sqrt {4}}=&2\\100&&{\sqrt {100}}=&10\\0,01&&{\sqrt {0,01}}=&0,1\\250\,000&&{\sqrt {250\,000}}=&500\end{matrix}}}
Arrels cúbiques
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Presentació:
a
3
{\displaystyle {\sqrt[{3}]{a}}}
arrel cúbica de a .
Definició: L'arrel cúbica de a és un valor real r , que compleix que
r
3
=
a
.
{\displaystyle r^{3}=a.}
Observació:
R
→
∙
3
R
0
0
3
=
0
1
1
3
=
1
−
1
−
1
3
=
−
1
8
8
3
=
2
−
8
−
8
3
=
−
2
−
1000
−
1000
3
=
10
0
,
001
0
,
001
3
=
0
,
1
−
8000
−
8000
3
=
20
{\displaystyle {\begin{matrix}\mathbb {R} &\xrightarrow {\sqrt[{3}]{\,\,\bullet \,\,}} &\mathbb {R} \\0&&{\sqrt[{3}]{0}}=&0\\1&&{\sqrt[{3}]{1}}=&1\\-1&&{\sqrt[{3}]{-1}}=&-1\\8&&{\sqrt[{3}]{8}}=&2\\-8&&{\sqrt[{3}]{-8}}=&-2\\-1000&&{\sqrt[{3}]{-1000}}=&10\\0,001&&{\sqrt[{3}]{0,001}}=&0,1\\-8000&&{\sqrt[{3}]{-8000}}=&20\end{matrix}}}
Arrels en general
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Presentació:
a
n
{\displaystyle {\sqrt[{n}]{a}}}
arrel n-èsima de a .
Definició: L'arrel n-èsima de a és un valor real r si
r
n
=
a
.
{\displaystyle r^{n}=a.}
a
n
=
{
a
n
=
x
si
a
>
0
a
n
=
x
si
n
és imparell
∄
=
No existeix
a la resta de casos
{\displaystyle {\sqrt[{n}]{a}}={\begin{cases}{\sqrt[{n}]{a}}=x&{\text{ si }}a>0\\{\sqrt[{n}]{a}}=x&{\text{ si }}n{\text{ és imparell }}\\\nexists ={\text{ No existeix}}&{\text{ a la resta de casos }}\end{cases}}}
Observació: L'arrel n-èsima r , si existeix , té el mateix signe de a .
R
→
∙
n
R
a
x
{\displaystyle {\begin{matrix}\mathbb {R} &\xrightarrow {\sqrt[{n}]{\,\,\bullet \,\,}} &\mathbb {R} \\a&&x\end{matrix}}}
Exemples
1).
1
100
=
1
{\displaystyle {\sqrt[{100}]{1}}=1}
3).
10000
4
=
10
{\displaystyle {\sqrt[{4}]{10000}}=10}
4).
−
1
7
=
−
1
{\displaystyle {\sqrt[{7}]{-1}}=-1}
a
Índex
→
n
←
Radicand
↙
Radical
=
x
←
arrel
{\displaystyle {\sqrt[{{\text{Índex}}\rightarrow n}]{a}}_{\leftarrow {\text{Radicand}}}^{\swarrow {\text{Radical}}}=x\leftarrow {\text{arrel}}}
Exercicis
1).
−
27
3
=
{\displaystyle {\sqrt[{3}]{-27}}=}
2).
16
4
=
{\displaystyle {\sqrt[{4}]{16}}=}
3).
16
=
{\displaystyle {\sqrt {{\sqrt {16}}\;}}=}
4).
32
5
=
{\displaystyle {\sqrt[{5}]{32}}=}
5).
64
6
=
{\displaystyle {\sqrt[{6}]{64}}=}
6).
64
3
=
{\displaystyle {\sqrt[{3}]{{\sqrt {64}}\;}}=}
7).
100
000
000
8
=
{\displaystyle {\sqrt[{8}]{100\,000\,000}}=}
8).
−
125
3
=
{\displaystyle {\sqrt[{3}]{-125}}=}
9).
10
7
7
=
{\displaystyle {\sqrt[{7}]{10^{7}}}=}
10).
100
000
000
=
{\displaystyle {\sqrt {{\sqrt {{\sqrt {100\,000\,000}}\;}}\;}}=}
11).
0
,
01
=
{\displaystyle {\sqrt {0,01}}=}
12).
0
,
0001
=
{\displaystyle {\sqrt {0,0001}}=}
13).
64
000
000
3
=
{\displaystyle {\sqrt[{3}]{\sqrt {64\,000\,000}}}=}
14).
10
14
7
=
{\displaystyle {\sqrt[{7}]{10^{14}}}=}
Propietats
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Per poder treballar directament sobre les propietats suposarem que no es pot reduir l'ultim quadrat de
(
−
a
)
2
=
a
{\displaystyle {\sqrt {\,(-a)^{2}\,}}=a}
−
a
2
{\displaystyle {\sqrt {\,-a\,}}^{\,2}}
Notació
a
n
=
a
1
n
{\displaystyle {\sqrt[{n}]{\,a\,}}=a^{\frac {1}{n}}}
a
n
m
{\displaystyle {\sqrt[{n}]{\,a\,}}^{\,m}}
=
(
a
1
n
)
m
{\displaystyle =\left(a^{\frac {1}{n}}\right)^{m}}
=
a
m
n
{\displaystyle =a^{\frac {m}{n}}}
a
m
n
{\displaystyle {\sqrt[{n}]{\,\,a^{m}\,\,}}}
=
(
a
m
)
1
n
{\displaystyle =\left(a^{m}\right)^{\frac {1}{n}}}
=
a
m
n
{\displaystyle =a^{\frac {m}{n}}}
I la fracció
m
n
{\displaystyle {\tfrac {m}{n}}}
a
n
n
{\displaystyle {\sqrt[{n}]{\,a\,}}^{\,n}}
=
a
n
n
{\displaystyle ={\sqrt[{n}]{\,\,a^{n}\,\,}}}
=
a
n
n
=
a
1
=
a
.
{\displaystyle =a^{\frac {n}{n}}=a^{1}=a.}
a
m
n
{\displaystyle {\sqrt[{n}]{\,{\sqrt[{m}]{\,a\,}}\,}}}
=
(
a
1
m
)
1
n
{\displaystyle =\left(a^{\frac {1}{m}}\right)^{\frac {1}{n}}}
=
a
1
m
⋅
1
n
{\displaystyle =a^{{\frac {1}{m}}\cdot {\frac {1}{n}}}}
=
a
1
n
⋅
m
{\displaystyle =a^{\frac {1}{n\cdot m}}}
=
a
n
⋅
m
.
{\displaystyle ={\sqrt[{n\cdot m}]{\,a\,}}.}
Propietats
a
⋅
b
n
=
a
n
⋅
b
n
{\displaystyle {\sqrt[{n}]{a\cdot b}}={\sqrt[{n}]{a}}\cdot {\sqrt[{n}]{b}}}
(
a
⋅
b
)
1
n
=
a
1
n
⋅
b
1
n
{\displaystyle (a\cdot b)^{\frac {1}{n}}=a^{\frac {1}{n}}\cdot b^{\frac {1}{n}}}
a
n
⋅
a
m
=
a
n
⋅
m
m
+
n
{\displaystyle {\sqrt[{n}]{a}}\cdot {\sqrt[{m}]{a}}={\sqrt[{n\cdot m}]{\,\,a\,\,}}^{\,m+n}}
a
1
n
⋅
a
1
m
=
a
1
n
+
1
m
{\displaystyle a^{\frac {1}{n}}\cdot a^{\frac {1}{m}}=a^{{\frac {1}{n}}+{\frac {1}{m}}}}
a
b
n
=
a
n
b
n
{\displaystyle {\sqrt[{n}]{\frac {a}{b}}}={\frac {\sqrt[{n}]{a}}{\sqrt[{n}]{b}}}}
(
a
b
)
1
n
=
a
1
n
b
1
n
{\displaystyle \left({\frac {a}{b}}\right)^{\frac {1}{n}}={\frac {a^{\frac {1}{n}}}{b^{\frac {1}{n}}}}}
a
n
a
m
=
a
n
⋅
m
m
−
n
{\displaystyle {\frac {\sqrt[{n}]{a}}{\sqrt[{m}]{a}}}={\sqrt[{n\cdot m}]{\,\,a\,\,}}^{\,m-n}}
a
1
n
a
1
m
=
a
1
n
−
1
m
{\displaystyle {\frac {a^{\frac {1}{n}}}{a^{\frac {1}{m}}}}=a^{{\frac {1}{n}}-{\frac {1}{m}}}}
![{\displaystyle {\sqrt[{3}]{a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/895424fc79dfd221f984d973ac95ca277bd0e60c)
![{\displaystyle {\begin{matrix}\mathbb {R} &\xrightarrow {\sqrt[{3}]{\,\,\bullet \,\,}} &\mathbb {R} \\0&&{\sqrt[{3}]{0}}=&0\\1&&{\sqrt[{3}]{1}}=&1\\-1&&{\sqrt[{3}]{-1}}=&-1\\8&&{\sqrt[{3}]{8}}=&2\\-8&&{\sqrt[{3}]{-8}}=&-2\\-1000&&{\sqrt[{3}]{-1000}}=&10\\0,001&&{\sqrt[{3}]{0,001}}=&0,1\\-8000&&{\sqrt[{3}]{-8000}}=&20\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b0b60bc2d430bbb69f768603873a78dee8bdf4c)
![{\displaystyle {\sqrt[{n}]{a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7873203eb76042fcd24056c553de8c86054a2df)
![{\displaystyle {\sqrt[{n}]{a}}={\begin{cases}{\sqrt[{n}]{a}}=x&{\text{ si }}a>0\\{\sqrt[{n}]{a}}=x&{\text{ si }}n{\text{ és imparell }}\\\nexists ={\text{ No existeix}}&{\text{ a la resta de casos }}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27556509718b5b5c3da974414efbf78664e1ffd9)
![{\displaystyle {\begin{matrix}\mathbb {R} &\xrightarrow {\sqrt[{n}]{\,\,\bullet \,\,}} &\mathbb {R} \\a&&x\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ea1d45588cf902a445128c88d693306350e9f1)
![{\displaystyle {\sqrt[{100}]{1}}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df4b467337d50d6383f284ab0068664ef18e71fc)
![{\displaystyle {\sqrt[{4}]{10000}}=10}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c89df78da693d95b0c0c3accc97362a0ac08575)
![{\displaystyle {\sqrt[{7}]{-1}}=-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2fd9386ac4d0f48239c6fc7e625ea50715964bf)
![{\displaystyle {\sqrt[{{\text{Índex}}\rightarrow n}]{a}}_{\leftarrow {\text{Radicand}}}^{\swarrow {\text{Radical}}}=x\leftarrow {\text{arrel}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4e0a460dc48e21df95682d099abb25cb2d73a60)
![{\displaystyle {\sqrt[{3}]{-27}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e47ace56267c75e6ef28d5411514a8e5b98d9d6d)
![{\displaystyle {\sqrt[{4}]{16}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6331cf5be42bf2783010a33626fec15c820e9d6f)
![{\displaystyle {\sqrt[{5}]{32}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ce72b6e3ae796485ed34576d5832ee49c66968f)
![{\displaystyle {\sqrt[{6}]{64}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2196bc9f4dd923b65feda3113114b4d95dfd43a7)
![{\displaystyle {\sqrt[{3}]{{\sqrt {64}}\;}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d9b4038a6f72d730f62cf8b4b077e0fe497e83e)
![{\displaystyle {\sqrt[{8}]{100\,000\,000}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df869b0f1bd462e4ae5f42e397023299a717d342)
![{\displaystyle {\sqrt[{3}]{-125}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe04f3ead568b943ffb404da0a1937ce65c63083)
![{\displaystyle {\sqrt[{7}]{10^{7}}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea0ca884a36d42c8605e69e1c28fa4d0602208fb)
![{\displaystyle {\sqrt[{3}]{\sqrt {64\,000\,000}}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8639f5884c678ed1d0bdf01ef77064158070e1cc)
![{\displaystyle {\sqrt[{7}]{10^{14}}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/244499af180aeafcff0ed48b90934513ae805120)
![{\displaystyle {\sqrt[{n}]{\,a\,}}=a^{\frac {1}{n}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef61ee4e1542e549756b5c15f36a15f2e1eaa735)
![{\displaystyle {\sqrt[{n}]{\,a\,}}^{\,m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37a14f7837ce6351a26528a8ddd3b8fd0d1d997a)
![{\displaystyle {\sqrt[{n}]{\,\,a^{m}\,\,}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5e67770be2a6136f2018c682247165558fb773e)
![{\displaystyle {\sqrt[{n}]{\,a\,}}^{\,n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49cb1284e9e9341a8a11d2ac63881267f62ab5ad)
![{\displaystyle ={\sqrt[{n}]{\,\,a^{n}\,\,}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e53370791866c33b54d152b66bffd9af5916d789)
![{\displaystyle {\sqrt[{n}]{\,{\sqrt[{m}]{\,a\,}}\,}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d8346e3a2dc0396fe6d65ef065b663856791e0e)
![{\displaystyle ={\sqrt[{n\cdot m}]{\,a\,}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce89cfa41de885807b704d0f94c7908ed66214e3)
![{\displaystyle {\sqrt[{n}]{a\cdot b}}={\sqrt[{n}]{a}}\cdot {\sqrt[{n}]{b}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ecfa1618d56f60c6ce1ad62d984d02b33967c02)
![{\displaystyle {\sqrt[{n}]{a}}\cdot {\sqrt[{m}]{a}}={\sqrt[{n\cdot m}]{\,\,a\,\,}}^{\,m+n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1bcdd64315e9a1d1559ac09aafec15732cd1fe07)
![{\displaystyle {\sqrt[{n}]{\frac {a}{b}}}={\frac {\sqrt[{n}]{a}}{\sqrt[{n}]{b}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/372ddb7d13541806e35a6053ba614df98a87b655)
![{\displaystyle {\frac {\sqrt[{n}]{a}}{\sqrt[{m}]{a}}}={\sqrt[{n\cdot m}]{\,\,a\,\,}}^{\,m-n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/389a966cc4616c8e8dc08f8be5bef55ed4395c7a)
