Machten en wortels
§1 Kwadraten
Kwadraten, uitwerkingen ...........................................................................................................
Van het vierkant hiernaast is iedere zijde 4.
De oppervlakte van het vierkant is 4 × 4 = 16.
In plaats van 4 × 4 schrijf je ook wel 42.
Je spreekt dat uit als: vier kwadraat.
| 32 = 3 × 3 = 9 |
10² = 10 × 10 = 100 |
| 52 = 5 × 5 = 25 |
15² = 15 × 15 = 225 |
| 82 = 8 × 8 = 64 |
20² = 20 × 20 = 400 |
| 3² + 7 = 16 |
(2 × 5)² = 100 |
| 5² - 20 = 5 |
5² + 3² = 34 |
| 40 - 5² = 15 |
2² + 4·3 = 16 |
De oppervlakte van het vierkant is a × a = a2
De oppervlakte van de rechthoek is 3 × a = 3a
De totale oppervlakte is dus a² + 3a.
| a |
1 |
2 |
3 |
4 |
5 |
| oppervlakte |
4 |
10 |
18 |
28 |
40 |
| 5² = 25 |
21 - 4² = 5 |
| 12² = 144 |
5·2² = 20 |
| 8 + 4² = 24 |
( 8 - 2² ) = 4 |
| a |
1 |
2 |
3 |
4 |
5 |
| oppervlakte |
3 |
8 |
15 |
24 |
35 |
| z |
1 |
2 |
3 |
4 |
5 |
| oppervlakte |
16 |
25 |
36 |
49 |
64 |
| n |
n² |
| 1 |
1 |
| 2 |
4 |
| 3 |
9 |
| 4 |
16 |
| 5 |
25 |
| 6 |
36 |
| 7 |
49 |
| 8 |
64 |
| 9 |
81 |
| 10 |
100 |
| 11 |
121 |
| 12 |
144 |
| 13 |
169 |
| 14 |
196 |
| 15 |
225 |
| 16 |
256 |
| 17 |
289 |
| 18 |
324 |
| 19 |
361 |
| 20 |
400 |
§2 Wortels
Wortels, Uitwerkingen ...................................................................................................
De zijde van het vierkant is 4 , want 4 × 4 = 16.
Je zegt de wortel van 16 is 4.
Je schrijft: √16 = 4.
| √9 = 3 , want 3 × 3 = 9 |
√100 = 10 , want 10 × 10 = 100 |
| √25 = 5 , want 5 × 5 = 25 |
√64 = 8 , want 8 × 8 = 64 |
| √36 = 6 , want 6 × 6 = 36 |
√1 = 1 , want 1 × 1 = 1 |
√20 ≈ 4,47 m
6,25Hm2= 62500m2
√62500 =250 m
√12 ligt tussen √9 en √16 en dus ligt √12 tussen 3 en 4.
√20 ligt tussen √16 en √25 en dus ligt √20 tussen 4 en 5.
| √36 = 6 |
√256 = 16 |
| √49 = 7 |
√196 = 14 |
| √144 = 12 |
√169 = 13 |
| √5 ≈ 2,24 |
√8 ≈ 2,83 |
| √49 = 7 |
√40 ≈ 6,32 |
| √60 ≈ 7,75 |
√121 = 11 |
| \(\small \mathsf { { \sqrt{4^2 + 3^2}=}}\) \(\small{ \mathsf{ \sqrt{25}=5 } }\) |
|
\(\small \mathsf { { \sqrt{4^2} + \sqrt{3^2} = 4 + 3 =7 }}\) |
| |
|
|
| \(\small \mathsf { { \sqrt{(4 + 3)^2}= }}\) \(\small{ \mathsf{ \sqrt{7} \approx2,65 } }\) |
|
\(\small{ \mathsf{ \sqrt{64 \ - 4^2\ - 1^2}= }}\) \(\small{ \mathsf{ \sqrt{64 - 16 + 1} = \sqrt{ 49} = 7} }\) |
| |
|
|
| \(\small \mathsf { {1 + \sqrt{16} =1+4=5}}\) |
|
\(\small \mathsf { { (\sqrt{27})^2 =27}}\) |
|
82 - √400 = 64 - 20 = 44
|
6 + 5 x 2 : √4 - 32 = 6 + 10 : 2 - 9 =
= 6 + 5 - 9 = 2 |
| |
|
(√49 - 4)2 = (7 - 4)2 = (3)2 = 9
|
32 + (6 + 3 x 5) : √36 = 9 + (6 + 15) : 6 =
= 9 + (21) : 6 =
= 30 : 6 = 5
|
| |
|
|
14 - √81 + 22 x 3 = 14 - 9 + 4 x 3 =
= 14 - 9 + 12 =
= 5 + 12 = 17
|
7 + (11 - 4)2 - √49 = 7 + (7)2 - 7 =
= 7 + 49 - 7 =
= 56 - 7 = 49
|
| \(\small \mathsf { { \sqrt{9 \over 16} }}\) = \(\mathsf{ \small{ {3 \over 4} } }\) |
\(\small \mathsf { { \sqrt{4\over 9} + \sqrt{81\over 36} }}\)= \(\mathsf{ \small{ {2 \over 3} } }\)+\(\mathsf{ \small{ {9 \over 6} } }\)= \(\mathsf{ \small{ 2{1 \over 6} } }\) |
| |
|
| \(\small \mathsf { { \sqrt{49\over 25} }}\) = \(\mathsf{ \small{ {7 \over 5} } }\) = \(\mathsf{ \small{ 1{2 \over 5} } }\) |
\(\small \mathsf { { \sqrt{16 \over 9} +\sqrt {169 \over 36} }}\)= \(\mathsf{ \small{ {4 \over 3} } }\)+\(\mathsf{ \small{ {13 \over 6} } }\)= \(\mathsf{ \small{ 3{1 \over 2} } }\) |
| |
|
| \(\small \mathsf { {1 + \sqrt{144\over 121} }}\)= 1 + \(\mathsf{ \small{ {12 \over11} } }\) =\(\mathsf{ \small{ 2{1 \over 11} } }\) |
\(\small \mathsf { { \sqrt{256\over196}+ {2\over9} }}\)= \(\mathsf{ \small{ {16 \over 14} } }\)+\(\mathsf{ \small{ {2 \over 9} } }\)= \(\mathsf{ \small{ {86 \over 63}=1{23\over63} } }\) |
| |
|
§3 Machten
2H03.3 Uitwerkingen .........................................................................................................
Van deze kubus zijn alle ribben 5 lang.
De inhoud is dan 5 × 5 × 5 = 125
In plaats van 5 × 5 × 5 schrijf je ook: 5³
Dit spreek je uit als 'vijf-tot-de-derde'
| 4³ = 4 × 4 × 4 = .... |
10³ = 10 × 10 × 10 = 1000
|
| 34 = 3 × 3 × 3 × 3 = 81 |
8² = 8 × 8 = 64 |
| 25 = 2 × 2 × 2 × 2 × 2 = 32 |
14 = 1 ×1 × 1 × 1 = 1 |
|
Z (zijde)
|
1
|
2
|
3
|
4
|
5
|
6
|
|
I (inhoud)
|
1
|
8
|
27
|
64
|
125
|
216
|
|
10³ < 310
|
210 > 1000 |
| 24 = 4² |
1³ < 31 |
| 54 < 45 |
43 < 34 |
| 10³ = 1000 |
64 = 1296 |
| 7³ = 343 |
7² + 4³ = 49 + 64 = 113 |
| 11³ = 1331 |
3³ - 24 = 27 - 16 = 11 |
|
\(\mathsf{ \small{ \sqrt{4^3 + 3^4 } } }\) = \(\small{ \mathsf{ \sqrt{64 + 81 } } }\) = \(\small{ \mathsf{ \sqrt{145} } }\)
≈ 12,04
|
\(\small{ \mathsf{ \sqrt{ 6^2 + \sqrt{2^8}\ \ -3\ } } }\) = \(\small{ \mathsf{ \sqrt{ 36 + \sqrt{256}\ \ -3\ } } }\)
= \(\small{ \mathsf{ \sqrt{ 36 + 16 -3\ } } }\) = \(\small{ \mathsf{ \sqrt{49} } }\) = 7
|
|
\(\small{ \mathsf{ \sqrt{4^3} + \sqrt{3^4 } } }\) = \(\small{ \mathsf{ \sqrt{64} + \sqrt{81 } } }\) =
8 + 9 = 17
|
\(\small{ \mathsf{ 43 - \sqrt{2^3 + 17\ } +\ \sqrt{4 } } }\) = \(\small{ \mathsf{ 43 - \sqrt{8 + 17\ } +\ \sqrt{4 } } }\)
= \(\small{ \mathsf{ 43 - \sqrt{25} +\ \sqrt{4 } } }\) = 43 -25 + 2 = 20
|
|
\(\small{ \mathsf{( \sqrt{4^3 + 3^4 })^2 } }\) = \(\small{ \mathsf{ (\sqrt{145})^2 } }\) = 145
|
\(\small{ \mathsf{ \sqrt{8^3 + 5^4 + 7^2 } } }\) = \(\small{ \mathsf{ \sqrt{512 + 625 + 49 } } }\) =
\(\small{ \mathsf{ \sqrt{1186} } }\) ≈ 34,44
|
|
\(\small{ \mathsf{ (\sqrt{4^3})^3 + (\sqrt{3^4 })^2 } }\) =
\(\small{ \mathsf{ (\sqrt{64})^3 + (\sqrt{81})^2 } }\) =
8³ + 9² = 512 + 81 = 593
|
\(\small{ \mathsf{ \sqrt{2^8 - 4^4 } } }\) = \(\small{ \mathsf{ \sqrt{256 - 256 } } }\) = \(\small{ \mathsf{ \sqrt{0} } }\)= 0
|
|
L (lengte)
|
30
|
35
|
40
|
45
|
50
|
|
Inhoud
|
12000
|
27875
|
49000
|
76125
|
110000
|
|
33 : √81 + 5 = 27 : 9 + 5 = ...
3 + 5 = 8
|
6 x (32 - 52) = 6 x (32 - 25) = ...
= 6 x (7) = 42
|
|
51 – 3 x 24 = 51 - 3 x 16 = ....
51 - 48 = 3
|
8 – 14 : (√36 + 1) + 33 = 8 - 14 : (6 + 1) + 27 = ...
= 8 - 14 : (7) + 27 = ...
= 8 - 2 + 27 = 33
|
§4 Bijzondere machten
2H04.4 Uitwerkingen .........................................................................................................................
|
a. 45 = 1024
|
|
e. 4-2 = \(\mathsf{ \small{ 1 \over 4^2 } }\) = \(\mathsf{ \small{ 1 \over 16 } }\) |
| b. 91 = 9 |
|
f. 3-3 = \(\mathsf{ \small{ 1 \over 3^3 } }\) = \(\mathsf{ \small{ 1 \over 27 } }\) |
| c. 130 = 1 |
|
g. 0-2 KAN NIET! |
| d. 04 = 0 |
|
h. 01 = 0 |
| a. 32 × 32 = 81 |
|
e. 34 = 81 |
| b. 43 × 44 = 16.384 |
|
f. 47 = 16.384 |
| c. 22 × 27 = 512 |
|
g. 29 = 512 |
| d. 53 × 55 = 390.625 |
|
h. 58 = 390.625 |
De uitkomsten in beide kolommen zijn gelijk.
**
| a. 56 : 54 = 25 |
|
e. 52 = 25 |
| b. 64 : 63 = 6 |
|
f. 61 = 6 |
| c. 47 : 44 = 64 |
|
g. 43 = 64 |
| d. 37 : 33 = 81 |
|
h. 34 = 81 |
De antwoorden in beide kolommen zijn gelijk
**
0-2 betekent \(\mathsf{ \small{ 1 \over 0^2 } }\) = \(\mathsf{ \small{ 1 \over 0} }\) Maar dat kan niet, want je mag/kan niet delen door nul!
Daarom bestaat 0-2 niet.
§5 Bewerkingen met nagatieve getallen
§6 Negatieve getallen en machten
2H03.6 Uitwerkingen ..........................................................................................................
| (-3)2 = -3 × -3 = 9 |
-32 = - 3 × 3 = -9 |
| (-3)3 = -3 × -3 × -3 = -27 |
-33 = - 3 × 3 × 3 = -27 |
| (-2)3 = -2 × -2 × -2 = -8 |
-23 = -2 × 2 × 2 = -8 |
| (-4)2 = -4 × -4 = 16 |
-42 = -4 × 4 = -16 |
| (-5)4 = -5 × -5 × -5 × -5 = 625 |
-54 = -5 × 5 × 5 × 5 = -625 |
| 5-2 = \(\color{ red} {\mathsf{ \small{ 1 \over 5^2} } } \) = \(\color{ red} {\mathsf{ \small{ 1 \over 25} } }\) |
(-3)0 = 1 |
|
(4 + 2)2 + 32 =
62 + 9 =
36 + 9 = 45
|
(42 - 19)3 - 42 =
(16 - 19)3 - 16 =
(-3)3 - 16 =
-27 - 16 = -43
|
|
4 + 23 : 4 =
4 + 8 : 4 =
4 + 2 = 6
|
-22 × -32 : 6 =
-4 × -9 : 6 =
36 : 6 = 6
|
|
\(\small{ \mathsf{ -6\ +3^2- \sqrt{5^2 - 3^2} } }\)
\(\color{red}{\small{ \mathsf{ -6\ +9- \sqrt{25 - 9} } }}\)
\(\color{red}{\small{ \mathsf{ -6\ +9- \sqrt{16} } }}\)
\(\color{red}{\small{ \mathsf{ -6\ +9-5 } } }\) = \(\color{red}{\small{ \mathsf{ -2 } }}\)
|
\(\small{ \mathsf{ -\sqrt{13^2 - 12^2}-3^3 } }\)
\(\color{red}{ \small{ \mathsf{ -\sqrt{169 - 144}-27 } } }\)
\(\color{red}{ \small{ \mathsf{ -\sqrt{25}-27 } } }\)
\(\color{red}{ \small{ \mathsf{ -5-27 } } }\) = \(\color{red}{ \small{ \mathsf{ -32 } } }\)
|
|
\(\small{ \mathsf{ ( \sqrt{5^2 - 3^2})^3:8-7 } }\) =
\(\color{red}{ \small{ \mathsf{ ( \sqrt{25 - 9})^3:8-7 } } }\) =
\(\color{red}{ \small{ \mathsf{ ( \sqrt{16})^3:8-7 } } }\) =
\(\color{red}{ \small{ \mathsf{ (4)^3:8-7 } } }\) =
\(\color{red}{ \small{ \mathsf{ 64:8-7 } } }\) =
\(\color{red}{ \small{ \mathsf{ 8-7 } } }\) = \(\color{red}{ \small{ \mathsf{ 1 } } }\)
|
\(\small{ \mathsf{ - \sqrt{11+(-3)^2+5} } }\) =
\(\color{red}{ \small{ \mathsf{ - \sqrt{11+ 9 +5} } } }\) =
\(\color{red}{ \small{ \mathsf{ - \sqrt{25} } } }\)= \(\color{red}{ \small{ \mathsf{ - 5 } } }\)
|
| (52 -32)3 : 82 - 72 = 15 |
92 + 8 × 9 : 18 = 85 |
| (63 + 3) - 42 = 203 |
33 : 3 + 2 × (10 - 12)2 = 17 |
| 5 × -32 + (22 + 5)2 = 36 |
19 + (-5)3 + 132 - 73 = -280 |
| 5-2 × 103 : 8 + 92 = 86 |
-42 × 2-3 + 42 × -23 = -130 |
D-toets
Herhalingsopgaven
Extra opgaven
2H03.E-I Uitwerkingen .............................................................................................
| 3² = 9 |
11² = 121 |
| 6² = 36 |
20² = 400 |
| 82 = 64 |
14² = 196 |
| 2² = 4 |
13² = 169 |
| 3² + 9 = 18 |
6 × 5 + 3² = 39 |
| 5² - 6 = 19 |
6² : 3 = 12 |
| 4² + 9² = 97 |
(3 + 4)² = 49 |
| 10² + 7² = 149 |
(8 - 4)² = 16 |
| √36 = 6 |
√25 = 5 |
| √49 = 7 |
√64 = 8 |
| √81 = 9 |
√1 = 1 |
| √121 = 11 |
√144 = 12 |
| 5³ + 24 = 141 |
√121 + 7³ = 354 |
| 6² + 54 = 661 |
6³ - √25 = 211 |
| 9² - 34 = 0 |
(4 + 10)² + √16 = 2000 |
| 44 - 7² = 207 |
√81 × √81 = 81 |
- oppervlakte = (5 + a)²
-
| a |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
| oppervlakte |
36 |
49 |
64 |
81 |
100 |
121 |
144 |
169 |
|
9² > √100
81 > 10
|
(5 + 7)² < 5 × 7²
144 < 245
|
|
56 > 8³
15625 > 512
|
12² > √121
144 > 121
|
|
√49 < 3²
7 < 9
|
64 > (6 + 6)²
1296 > 144
|
|
10.000 > 210
10.000 > 1024
|
4² > √169
16 > 13
|
Extra: rekenvolgorde
Extra: Rekenvolgorde Uitwerkingen ..................................................................................
5 + 2 × 4 = 5 + 8 = 13
(10 − 2) × 3 = 8 × 3 = 24
5 × 5 + 3 = 25 + 3 = 28
20 − 8 × 2 = 20 − 16 = 4
52 × (4 + 5) = 52 × 9 = € 468,-
|
31 + 2 × 5 = 31 + 10 = 41
|
|
6 + 4 × 8 + 2 = 6 + 32 + 2 = 40 |
| 3 × 4 − 1 = 12 − 1 = 11 |
|
(6 + 4) × 8 + 2 = 10 × 8 + 2 = 82 |
| (12 – 3) × 2 = 9 × 2 = 18 |
|
6 + 4 × (8 + 2) = 6 + 4 × 10 = 46 |
| 6 × (3 + 2) = 6 × 5 = 30 |
|
(6 + 4) × (8 – 6) = 10 × 2 = 20 |
4 × (2,00 + 7,50 + 3,20) = 4 × 12,70 = € 50,80
Flippo 1: 58 oplossingen
|
1: (2*(2+6))+8 2: (2+(2*8))+6 3: 2+((2*8)+6) 4: (2+6)+(2*8)
5: 2+(6+(2*8)) 6: ((2+6)*2)+8 7: (2-6)*(2-8) 8: (2*(6+2))+8
9: 2*((6-2)+8) 10: 2*(6-(2-8)) 11: (2+6)+(8*2) 12: 2+(6+(8*2))
13: 2*((6+8)-2) 14: 2*(6+(8-2)) 15: (2+(8*2))+6 16: 2+((8*2)+6)
17: (2-8)*(2-6) 18: ((2*8)+2)+6 19: (2*8)+(2+6) 20: 2*((8-2)+6)
21: 2*(8-(2-6)) 22: ((2*8)+6)+2 23: (2*8)+(6+2) 24: 2*((8+6)-2)
25: 2*(8+(6-2)) 26: (6+2)+(2*8) 27: 6+(2+(2*8)) 28: ((6+2)*2)+8
29: (6+2)+(8*2) 30: 6+(2+(8*2)) 31: (6+(2*8))+2 32: 6+((2*8)+2)
33: ((6-2)+8)*2 34: (6-(2-8))*2 35: (6-2)*(8-2) 36: ((6+8)-2)*2
37: (6+(8-2))*2 38: (6+(8*2))+2 39: 6+((8*2)+2) 40: 6*(8-(2+2))
41: 6*((8-2)-2) 42: 6*(8-(2*2)) 43: 8+(2*(2+6)) 44: (8-(2+2))*6
45: ((8-2)-2)*6 46: (8-(2*2))*6 47: ((8*2)+2)+6 48: (8*2)+(2+6)
49: 8+((2+6)*2) 50: 8+(2*(6+2)) 51: ((8-2)+6)*2 52: (8-(2-6))*2
53: (8-2)*(6-2) 54: ((8*2)+6)+2 55: (8*2)+(6+2) 56: 8+((6+2)*2)
57: ((8+6)-2)*2 58: (8+(6-2))*2
|
Flippo 2: 42 oplossingen:
| |
1: ((2+4)*5)-6 2: (2*(4+5))+6 3: (2*(5+4))+6 4: ((4+2)*5)-6
5: 4-((2-6)*5) 6: ((4+5)*2)+6 7: 4-(5*(2-6)) 8: ((4*5)-2)+6
9: (4*5)-(2-6) 10: 4+(5*(6-2)) 11: ((4*5)+6)-2 12: (4*5)+(6-2)
13: 4+((6-2)*5) 14: (5*(2+4))-6 15: ((5+4)*2)+6 16: (5*(4+2))-6
17: ((5*4)-2)+6 18: (5*4)-(2-6) 19: ((5*4)+6)-2 20: (5*4)+(6-2)
21: (5*(6-2))+4 22: (5*6)-(2+4) 23: ((5*6)-2)-4 24: (5*6)-(4+2)
25: ((5*6)-4)-2 26: 6+(2*(4+5)) 27: (6-2)+(4*5) 28: 6-(2-(4*5))
29: 6+(2*(5+4)) 30: (6-2)+(5*4) 31: 6-(2-(5*4)) 32: ((6-2)*5)+4
33: 6+((4+5)*2) 34: (6+(4*5))-2 35: 6+((4*5)-2) 36: (6*5)-(2+4)
37: ((6*5)-2)-4 38: 6+((5+4)*2) 39: (6+(5*4))-2 40: 6+((5*4)-2)
41: (6*5)-(4+2) 42: ((6*5)-4)-2
|
Flippo 3: 4 oplossingen
| |
1: (2*(5+8))-2 2: (2*(8+5))-2 3: ((5+8)*2)-2 4: ((8+5)*2)-2 |
