Getal & Ruimte (13e editie) - vwo wiskunde B

'Wortels vereenvoudigen'.

2 vwo 5.3 Wortels herleiden

Wortels vereenvoudigen (5)

opgave 1

Herleid.

2p

a

\(\sqrt{50} + \sqrt{200}\)

Optellen (5)
0085 - Wortels vereenvoudigen - basis - 0ms

a

\(\sqrt{50} + \sqrt{200} = \sqrt{25} ⋅ \sqrt{2} + \sqrt{100} ⋅ \sqrt{2} = 5 \sqrt{2} + 10 \sqrt{2} \text{.}\)

1p

○

\(5 \sqrt{2} + 10 \sqrt{2} = 15 \sqrt{2} \text{.}\)

1p

1p

b

\(\sqrt{20}\)

FactorVoorWortelteken (1)
0086 - Wortels vereenvoudigen - basis - 0ms

b

\(\sqrt{20} = \sqrt{4} ⋅ \sqrt{5} = 2 \sqrt{5} \text{.}\)

1p

1p

c

\(-6 \sqrt{20}\)

FactorVoorWortelteken (2)
0087 - Wortels vereenvoudigen - basis - 0ms

c

\(-6 \sqrt{20} = -6 ⋅ \sqrt{4} ⋅ \sqrt{5} = -6 ⋅ 2 ⋅ \sqrt{5} = -12 \sqrt{5} \text{.}\)

1p

2p

d

\(7 \sqrt{27} - 2 \sqrt{300}\)

Optellen (6)
0088 - Wortels vereenvoudigen - basis - 0ms

d

\(7 \sqrt{27} - 2 \sqrt{300} = 7 ⋅ \sqrt{9} ⋅ \sqrt{3} - 2 ⋅ \sqrt{100} ⋅ \sqrt{3} \text{.}\)

1p

○

\(7 ⋅ 3 ⋅ \sqrt{3} - 2 ⋅ 10 ⋅ \sqrt{3} = 21 \sqrt{3} - 20 \sqrt{3} = 1 \sqrt{3} \text{.}\)

1p

opgave 2

Herleid.

1p

\(\sqrt{\frac{25}{36}}\)

BreukInWortel (1)
008b - Wortels vereenvoudigen - basis - 25ms

○

\(\sqrt{\frac{25}{36}} = {\sqrt{25} \over \sqrt{36}} = \frac{5}{6} \text{.}\)

1p
3 vwo 5.5 Wortels herleiden

Wortels vereenvoudigen (6)

opgave 1

Herleid.

1p

a

\({5 \over 8 \sqrt{5}}\)

WortelInNoemer
0089 - Wortels vereenvoudigen - basis - 0ms

a

\({5 \over 8 \sqrt{5}} = {5 \over 8 \sqrt{5}} ⋅ {\sqrt{5} \over \sqrt{5}} = {5 \sqrt{5} \over 8 ⋅ 5} = \frac{1}{8} \sqrt{5} \text{.}\)

1p

1p

b

\(\sqrt{\frac{12}{25}}\)

BreukInWortel (2)
008c - Wortels vereenvoudigen - basis - 1ms

b

\(\sqrt{\frac{12}{25}} = {\sqrt{12} \over \sqrt{25}} = {\sqrt{12} \over 5} = \frac{1}{5} \sqrt{12} = \frac{1}{5} ⋅ 2 ⋅ \sqrt{3} = \frac{2}{5} \sqrt{3} \text{.}\)

1p

1p

c

\(\sqrt{3\frac{1}{5}}\)

BreukInWortel (3)
008d - Wortels vereenvoudigen - basis - 0ms

c

\(\sqrt{3\frac{1}{5}} = \sqrt{\frac{16}{5}} = {\sqrt{16} \over \sqrt{5}} = {4 \over \sqrt{5}} ⋅ {\sqrt{5} \over \sqrt{5}} = {4 \sqrt{5} \over 5} = \frac{4}{5} \sqrt{5} \text{.}\)

1p

1p

d

\(\sqrt{1\frac{1}{7}}\)

BreukInWortel (4)
008e - Wortels vereenvoudigen - basis - 0ms

d

\(\sqrt{1\frac{1}{7}} = \sqrt{\frac{8}{7}} = {\sqrt{8} \over \sqrt{7}} ⋅ {\sqrt{7} \over \sqrt{7}} = {\sqrt{56} \over 7} = \frac{1}{7} \sqrt{56} = \frac{1}{7} ⋅ 2 ⋅ \sqrt{14} = \frac{2}{7} \sqrt{14} \text{.}\)

1p

opgave 2

Herleid.

1p

a

\({12 \sqrt{56} \over 2 \sqrt{7}}\)

Delen (4)
00dc - Wortels vereenvoudigen - basis - 1ms

a

\({12 \sqrt{56} \over 2 \sqrt{7}} = {12 \over 2} ⋅ {\sqrt{56} \over \sqrt{7}} = 6 \sqrt{8} = 6 ⋅ \sqrt{4} ⋅ \sqrt{2} = 6 ⋅ 2 ⋅ \sqrt{2} = 12 \sqrt{2}\)

1p

1p

b

\(4 \sqrt{3} ⋅ 3 \sqrt{6}\)

Vermenigvuldigen (5)
00dd - Wortels vereenvoudigen - basis - 2ms - data pool: #22 (2ms)

b

\(4 \sqrt{3} ⋅ 3 \sqrt{6} = 12 \sqrt{18} = 12 ⋅ \sqrt{9} ⋅ \sqrt{2} = 12 ⋅ 3 ⋅ \sqrt{2} = 36 \sqrt{2}\)

1p
vwo wiskunde B 3.3 Vergelijkingen in de meetkunde

Wortels vereenvoudigen (2)

opgave 1

Herleid.

1p

a

\({-3 \over 2 + \sqrt{6}}\)

SomInNoemer (1)
00r3 - Wortels vereenvoudigen - basis - 0ms

a

\({-3 \over 2 + \sqrt{6}} = {-3 \over 2 + \sqrt{6}} ⋅ {2 - \sqrt{6} \over 2 - \sqrt{6}}\)
\(\text{} = {-3 (2 + \sqrt{6}) \over 4 - 6}\)
\(\text{} = 1\frac{1}{2} (2 + \sqrt{6})\)
\(\text{} = 3 + 1\frac{1}{2} \sqrt{6}\)

1p

1p

b

\({5 \sqrt{6} \over \sqrt{2} - \sqrt{3}}\)

SomInNoemer (2)
00r4 - Wortels vereenvoudigen - basis - 0ms

b

\({5 \sqrt{6} \over \sqrt{2} - \sqrt{3}} = {5 \sqrt{6} \over \sqrt{2} - \sqrt{3}} ⋅ {\sqrt{2} + \sqrt{3} \over \sqrt{2} + \sqrt{3}}\)
\(\text{} = {5 \sqrt{6} (\sqrt{2} + \sqrt{3}) \over 2 - 3}\)
\(\text{} = -5 \sqrt{6} (\sqrt{2} + \sqrt{3})\)
\(\text{} = -5 \sqrt{12} - 5 \sqrt{18}\)
\(\text{} = -10 \sqrt{3} - 15 \sqrt{2}\)

1p
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