MATEMATIKA
Úvod
Číselné obory
Číselné obory - př.
Poměr a úměra
Procenta
Množiny
základní pojmy
intervaly
absolutní hodnota čísla
Logika
výroky
kvantifikované výroky
Mocniny
a odmocniny
mocniny s přirozeným exponentem
mocniny s celočíselným exponentem
mocniny s racionálním exponentem
n-tá odmocnina
početní výkony s odmocninami
zápis čísla ve tvaru a.10
n
Algebraické výrazy
výrazy a mnohočleny
úpravy výrazů
lomené výrazy
Funkce
vlastnosti
lineární
kvadratické
mocninné
exponenciální
logaritmické
Rovnice
lineární
kvadratické
iracionální
exponenciální
goniometrické
logaritmus
logaritmické rovnice
Nerovnice
lineární
kvadratické
exponenciální
logaritmické
Soustavy rovnic
a nerovnic
Soustavy lineárních rovnic
Soustavy lineárních a kvadratických rovnic
Goniometrie
goniometrické funkce
Planimetrie
shodnost a podobnost trojúhelníků
Euklidovy věty
rovinné obrazce
Trigonometrie
pravoúhlý trojúhelník
obecný trojúhelník
Komplexní čísla
algebraický tvar
goniometrický tvar
Moivreova věta
kvadratické rovnice v oboru C
Posloupnosti
aritmetická posloupnost
geometrická posloupnost
Analytická
geometrie v rovině
vektory
přímka
vzájemná poloha přímek a bodů
kružnice
elipsa
parabola
hyperbola
Kombinatorika
Variace
Permutace
Kombinace
Kombinační čísla a jejich vlastnosti
Binomická věta
Pravděpodobnost
příklady
Aplikace matematiky
bankovnictví
investice
statistika
N-tá odmocnina - příklady
Příklad 1
Zjednoduš:
a
4
6
a
2
3
b
16
20
b
4
5
c
12
8
c
3
d
15
9
d
5
3
121
6
11
3
32
15
2
3
16
2
4
4
8
⋅
3
3
3
6
(
x
4
3
)
5
x
6
⋅
x
2
3
(
x
3
⋅
x
4
)
4
x
13
a
16
b
8
c
4
4
a
4
b
2
c
27
4
6
9
Řešení
a
4
6
=
a
2
⋅
2
3
⋅
2
=
a
2
3
b
16
20
=
b
4
⋅
4
5
⋅
4
=
b
4
5
c
12
8
=
c
3
⋅
4
2
⋅
4
=
c
3
d
15
9
=
d
5
⋅
3
3
⋅
3
=
d
5
3
121
6
=
11
2
2
⋅
3
=
11
3
32
15
=
2
5
3
⋅
5
=
2
3
16
2
4
=
(
2
4
)
2
4
=
2
4
⋅
2
4
=
2
2
=
4
8
⋅
3
3
3
=
8
3
⋅
3
3
3
=
2
3
3
⋅
3
=
2
⋅
3
=
6
(
x
4
3
)
5
=
x
4
⋅
5
3
=
x
20
3
=
x
18
⋅
x
2
3
=
x
18
3
⋅
x
2
3
=
x
6
⋅
x
2
3
(
x
3
⋅
x
4
)
4
=
x
12
⋅
x
=
x
13
a
16
b
8
c
4
4
=
a
16
4
⋅
b
8
4
⋅
c
4
4
=
a
4
b
2
c
27
4
6
=
(
3
3
)
4
6
=
3
12
6
=
3
2
⋅
6
6
=
3
2
=
9
Příklad 2
Zjednoduš:
3
⋅
15
⋅
5
1
2
15
b
3
2
⋅
b
b
2
x
⋅
y
1
2
⋅
x
3
2
⋅
y
3
(
x
y
)
2
10
⋅
12
⋅
30
60
50
3
⋅
20
3
10
2
a
⋅
2
b
⋅
a
b
2
a
b
(
2
3
−
3
2
+
4
6
)
⋅
6
6
(
2
−
3
+
4
)
a
(
a
+
b
)
−
b
(
a
−
b
)
a
+
b
Řešení
3
⋅
15
⋅
5
1
2
=
=
3
⋅
15
⋅
5
=
=
3
⋅
15
⋅
5
=
=
225
=
15
b
3
2
⋅
b
=
=
b
3
⋅
b
=
=
b
3
⋅
b
=
=
b
4
=
=
b
2
x
⋅
y
1
2
⋅
x
3
2
⋅
y
3
=
=
x
⋅
x
3
⋅
y
⋅
y
3
=
=
x
4
⋅
y
4
=
=
x
2
⋅
y
2
=
(
x
y
)
2
10
⋅
12
⋅
30
=
=
10
⋅
12
⋅
30
=
=
3600
=
=
60
50
3
⋅
20
3
=
=
50
⋅
20
3
=
=
1000
3
=
=
10
2
a
⋅
2
b
⋅
a
b
=
=
2
a
⋅
2
b
⋅
a
b
=
=
4
a
2
b
2
=
=
2
a
b
(
2
3
−
3
2
+
4
6
)
⋅
6
=
=
2
18
−
3
12
+
4
36
=
=
2
9
⋅
2
−
3
4
⋅
3
+
4
⋅
6
=
=
2
⋅
3
2
−
3
⋅
2
3
+
24
=
=
6
2
−
6
3
+
24
=
=
6
(
2
−
3
+
4
)
a
(
a
+
b
)
−
b
(
a
−
b
)
=
=
a
+
a
b
−
a
b
+
b
=
=
a
+
b
Příklad 3
Zjednoduš:
52
13
2
16
a
b
2
3
2
a
2
b
3
2
⋅
a
b
3
84
1
2
⋅
21
−
1
2
x
3
5
⋅
x
−
8
5
1
x
(
3
x
y
5
)
−
1
2
⋅
27
x
3
y
3
x
y
2
32
x
y
2
:
(
2
x
2
y
)
1
2
4
⋅
y
x
16
81
4
2
3
56
3
7
3
2
16
3
:
2
3
2
60
:
15
2
3
⋅
15
5
3
2
⋅
12
6
⋅
24
⋅
2
3
1
3
2
x
y
3
⋅
y
3
x
3
⋅
4
x
2
y
3
⋅
y
2
9
x
3
2
3
⋅
x
y
3
Řešení
52
13
=
52
13
=
4
=
2
16
a
b
2
3
2
a
2
b
3
=
16
a
b
2
2
a
2
b
3
=
8
b
a
3
=
8
3
⋅
a
b
3
=
2
⋅
a
b
3
84
1
2
⋅
21
−
1
=
84
⋅
1
21
=
84
21
=
4
=
2
x
3
5
⋅
x
−
8
5
=
x
3
5
⋅
1
x
8
5
=
x
3
x
8
5
=
1
x
5
5
=
1
x
(
3
x
y
5
)
−
1
2
⋅
27
x
3
y
=
27
x
3
y
(
3
x
y
5
)
1
2
=
27
x
3
y
3
x
y
5
=
27
x
3
y
3
x
y
5
=
9
x
2
y
4
=
3
x
y
2
32
x
y
2
:
(
2
x
2
y
)
1
2
=
32
x
y
2
(
2
x
2
y
)
1
2
=
32
x
y
2
2
x
2
y
=
16
y
x
=
4
⋅
y
x
16
81
4
=
16
4
81
4
=
2
4
4
3
4
4
=
2
3
56
3
7
3
=
56
7
3
=
8
3
=
2
16
3
:
2
3
=
16
3
2
3
=
16
2
3
=
8
3
=
2
60
:
15
=
60
15
=
60
15
=
4
=
2
3
⋅
15
5
=
45
5
=
45
5
=
9
=
3
2
⋅
12
6
⋅
24
⋅
2
3
=
2
⋅
12
⋅
2
6
⋅
24
⋅
3
=
1
9
=
1
3
2
x
y
3
⋅
y
3
x
3
⋅
4
x
2
y
3
⋅
y
2
9
x
3
=
2
x
⋅
y
⋅
4
x
2
⋅
y
2
y
⋅
3
x
⋅
y
⋅
9
x
3
=
8
x
3
y
3
27
x
2
y
2
3
=
=
8
27
3
⋅
x
3
y
3
x
2
y
2
3
=
2
3
⋅
x
y
3
Příklad 4
Zjednoduš:
25
5
5
5
1
81
3
1
9
3
a
4
5
4
a
5
9
3
⋅
25
3
15
3
a
⋅
b
3
3
a
3
b
1
a
x
9
4
3
1
16
4
2
⋅
x
3
4
5
3
5
6
27
4
3
3
4
Řešení
25
5
=
25
2
⋅
5
=
5
2
2
⋅
5
=
5
5
1
81
3
=
1
81
2
⋅
3
=
1
9
2
2
⋅
3
=
1
9
3
a
4
5
4
=
a
4
4
⋅
5
=
a
5
9
3
⋅
25
3
=
3
2
2
⋅
3
⋅
5
2
3
⋅
2
=
3
3
⋅
5
3
=
3
⋅
5
3
=
15
3
a
⋅
b
3
3
a
3
b
=
a
⋅
b
3
3
⋅
2
a
3
b
=
a
⋅
b
a
3
b
=
a
b
a
3
b
=
a
b
a
3
b
=
1
a
2
=
1
a
x
9
4
3
1
16
4
=
x
9
3
⋅
4
1
2
4
4
=
x
3
⋅
3
3
⋅
4
1
2
=
x
3
4
1
2
=
2
⋅
x
3
4
5
3
=
5
2
⋅
3
=
5
6
27
4
3
=
3
3
3
⋅
4
=
3
4
Příklad 5
Vypočítej:
5
+
3
3
−
(
2
5
−
3
5
−
3
)
4
3
+
2
5
5
(
2
+
3
)
−
4
(
3
−
2
)
−
3
(
3
2
−
3
)
4
3
8
a
+
5
x
−
7
a
+
4
a
−
6
x
−
3
a
+
x
2
a
a
(
a
+
b
)
−
a
(
a
−
b
)
−
a
b
+
b
a
2
b
a
(
3
+
2
)
⋅
(
12
−
8
)
2
(
3
7
+
2
)
⋅
(
3
7
−
2
)
61
(
2
+
5
−
3
)
⋅
(
2
+
5
+
3
)
4
+
2
10
(
3
−
2
)
⋅
(
3
+
2
)
1
(
3
x
−
2
y
)
2
9
x
−
12
x
y
+
4
y
(
3
−
5
)
2
+
(
1
+
5
)
2
−
(
2
5
−
1
)
2
−
1
3
−
2
3
+
2
+
3
+
2
3
−
2
10
a
+
b
a
−
b
−
a
−
b
a
+
b
4
a
b
a
2
−
b
2
3
−
5
4
⋅
3
+
9
⋅
3
−
25
⋅
3
−
10
3
5
6
−
7
24
+
3
54
−
2
96
+
2
150
2
6
2
(
3
+
6
)
+
3
(
2
+
6
)
−
6
(
2
+
3
)
2
6
(
3
5
+
2
10
)
2
85
+
60
2
(
6
+
2
)
3
20
2
+
12
6
Řešení
5
+
3
3
−
(
2
5
−
3
5
−
3
)
=
5
+
3
3
−
2
5
+
3
5
+
3
=
=
3
3
+
3
+
5
−
2
5
+
3
5
=
4
3
+
2
5
5
(
2
+
3
)
−
4
(
3
−
2
)
−
3
(
3
2
−
3
)
=
=
5
2
+
5
3
−
4
3
+
4
2
−
9
2
+
3
3
=
=
5
2
+
4
2
−
9
2
+
5
3
−
4
3
+
3
3
=
4
3
8
a
+
5
x
−
7
a
+
4
a
−
6
x
−
3
a
+
x
=
=
8
a
−
7
a
+
4
a
−
3
a
+
5
x
−
6
x
+
x
=
2
a
a
(
a
+
b
)
−
a
(
a
−
b
)
−
a
b
+
b
a
=
=
a
a
+
a
b
−
a
a
+
b
a
−
a
b
+
b
a
=
2
b
a
(
3
+
2
)
⋅
(
12
−
8
)
=
3
⋅
12
−
3
⋅
8
+
2
⋅
12
−
2
⋅
8
=
=
36
−
24
+
24
−
16
=
6
−
4
=
2
(
3
7
+
2
)
⋅
(
3
7
−
2
)
=
(
3
7
)
2
−
(
2
)
2
=
9
⋅
7
−
2
=
61
(
2
+
5
−
3
)
⋅
(
2
+
5
+
3
)
=
=
[
(
2
+
5
)
−
3
]
⋅
[
(
2
+
5
)
+
3
]
=
(
2
+
5
)
2
−
(
3
)
2
=
=
2
+
2
10
+
5
−
3
=
4
+
2
10
(
3
−
2
)
⋅
(
3
+
2
)
=
(
3
)
2
−
(
2
)
2
=
3
−
2
=
1
(
3
x
−
2
y
)
2
=
9
x
−
12
x
⋅
y
+
4
y
=
9
x
−
12
x
y
+
4
y
(
3
−
5
)
2
+
(
1
+
5
)
2
−
(
2
5
−
1
)
2
=
=
9
−
6
5
+
5
+
1
+
2
5
+
5
−
(
4
⋅
5
−
4
5
+
1
)
=
=
9
−
6
5
+
5
+
1
+
2
5
+
5
−
20
+
4
5
−
1
=
−
1
3
−
2
3
+
2
+
3
+
2
3
−
2
=
(
3
−
2
)
2
+
(
3
+
2
)
2
(
3
+
2
)
(
3
−
2
)
=
=
3
−
2
6
+
2
+
3
+
2
6
+
2
3
−
2
=
10
a
+
b
a
−
b
−
a
−
b
a
+
b
=
(
a
+
b
)
2
−
(
a
−
b
)
2
(
a
−
b
)
(
a
+
b
)
=
=
a
2
+
2
a
b
+
b
−
(
a
2
−
2
a
b
+
b
)
a
2
−
b
=
=
a
2
+
2
a
b
+
b
−
a
2
+
2
a
b
−
b
a
2
−
b
=
4
a
b
a
2
−
b
2
3
−
5
4
⋅
3
+
9
⋅
3
−
25
⋅
3
=
=
2
3
−
5
⋅
2
3
+
3
3
−
5
3
=
−
10
3
5
6
−
7
24
+
3
54
−
2
96
+
2
150
=
=
5
6
−
7
4
⋅
6
+
3
9
⋅
6
−
2
16
⋅
6
+
2
25
⋅
6
=
=
5
6
−
7
⋅
2
6
+
3
⋅
3
6
−
2
⋅
4
6
+
2
⋅
5
6
=
2
6
2
(
3
+
6
)
+
3
(
2
+
6
)
−
6
(
2
+
3
)
=
=
6
+
12
+
6
+
18
−
12
−
18
=
2
6
(
3
5
+
2
10
)
2
=
9
⋅
5
+
12
5
⋅
10
+
4
⋅
10
=
45
+
12
50
+
40
=
=
85
+
12
50
=
85
+
12
25
⋅
2
=
85
+
12
⋅
5
2
=
85
+
60
2
(
6
+
2
)
3
=
6
6
+
3
⋅
6
⋅
2
+
3
6
⋅
2
+
2
2
=
=
6
6
+
18
2
+
6
6
+
2
2
=
20
2
+
12
6
Příklad 6
Zjednoduš:
8
a
3
5
⋅
(
4
a
2
)
1
5
2
a
32
a
5
5
⋅
(
2
a
3
)
−
1
2
1
a
s
4
⋅
s
5
6
⋅
(
s
−
1
)
3
1
s
⋅
s
7
12
k
7
18
⋅
k
−
1
9
k
−
1
3
k
11
18
y
3
⋅
y
4
7
y
13
28
x
3
⋅
x
4
3
x
x
2
3
x
4
100
−
20
m
+
m
2
|
10
−
m
|
(
n
0
⋅
n
−
5
4
n
1
4
)
8
1
n
12
5
⋅
3
3
6
6
:
6
⋅
5
3
3
⋅
3
3
2
−
1
⋅
3
18
Řešení
8
a
3
5
⋅
(
4
a
2
)
1
5
=
8
a
3
5
⋅
4
a
2
5
=
32
a
5
5
=
2
5
a
5
5
=
2
a
32
a
5
5
⋅
(
2
a
3
)
−
1
2
=
32
a
5
5
(
2
a
3
)
1
2
=
32
a
5
10
2
a
3
=
2
5
a
5
2
⋅
5
2
a
3
=
2
a
2
a
3
=
=
2
a
2
a
3
=
1
a
2
=
1
a
s
4
⋅
s
5
6
⋅
(
s
−
1
)
3
=
s
4
⋅
s
5
6
⋅
s
−
3
=
s
3
4
⋅
3
⋅
s
5
⋅
2
6
⋅
2
⋅
1
s
3
=
=
s
3
⋅
s
10
12
⋅
1
s
1
s
=
1
s
⋅
s
13
12
⋅
1
s
6
2
⋅
6
=
1
s
⋅
s
13
s
6
12
=
1
s
⋅
s
7
12
k
7
18
⋅
k
−
1
9
k
−
1
3
=
k
7
18
⋅
k
−
1
⋅
2
9
⋅
2
k
−
1
⋅
6
3
⋅
6
=
k
7
18
⋅
k
−
2
18
k
−
6
18
=
k
7
⋅
k
−
2
18
k
−
6
18
=
k
5
k
−
6
18
=
k
11
18
y
3
⋅
y
4
7
=
y
3
⋅
4
4
⋅
y
4
7
=
y
12
⋅
y
4
7
=
y
13
28
x
3
⋅
x
4
3
x
⋅
x
2
3
=
x
3
⋅
4
4
⋅
x
4
3
x
3
3
⋅
x
2
3
=
x
12
⋅
x
4
3
x
3
⋅
x
2
3
=
x
13
12
x
5
6
=
x
13
12
x
5
⋅
2
6
⋅
2
=
x
13
x
10
12
=
=
x
3
12
=
x
4
100
−
20
m
+
m
2
=
(
10
−
m
)
2
=
|
10
−
m
|
(
n
0
⋅
n
−
5
4
n
1
4
)
8
=
(
n
−
5
4
n
4
)
8
=
(
n
−
5
n
4
)
8
=
(
1
n
6
4
)
8
=
1
n
48
4
=
1
n
12
5
⋅
3
3
6
6
:
6
⋅
5
3
3
⋅
3
3
=
5
3
3
⋅
3
3
6
6
⋅
3
2
⋅
3
3
6
2
⋅
5
3
=
5
3
⋅
3
3
6
6
⋅
3
2
⋅
3
3
6
2
⋅
5
3
=
=
5
3
⋅
3
3
6
3
3
6
⋅
3
3
6
6
2
⋅
5
6
=
5
3
⋅
3
6
3
3
6
⋅
3
3
6
2
⋅
5
6
=
5
3
⋅
3
6
3
18
⋅
3
3
⋅
3
6
2
⋅
3
⋅
5
3
6
⋅
3
=
=
5
3
⋅
3
⋅
3
9
6
3
⋅
6
6
⋅
5
3
18
=
3
10
6
9
18
=
3
10
2
9
⋅
3
9
18
=
3
2
9
18
=
1
2
⋅
3
18
=
2
−
1
⋅
3
18
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