UHU-Startseite►Mathematik►Jahrgangsstufen►9►Weiterentwicklung der Zahlvorstellung►Umgehen mit einfachen Wurzeltermen►(Lösungen)►
1.3


3
a) `sqrt 48 = sqrt (2*2*2*2*3) = 2*2*sqrt 3 = 4 sqrt 3`

h) `sqrt (2^3*5^2*7^3) = sqrt (2^2*5^2*7^2*2*7) = 2*5*7*sqrt(2*7) = 70 sqrt 14`




4
f) `27*sqrt (1/3) = sqrt(27^2/3) = sqrt(27*9) = sqrt 243`




9
a) `(3 sqrt 18 - 4 sqrt 10)(sqrt 10 + sqrt 2) = (3*3 sqrt 2 - 4 sqrt 10)(sqrt 10 + sqrt 2)`

` = 3*3 sqrt 2 * sqrt 10 + 3*3 sqrt 2 * sqrt 2 - 4 sqrt 10 * sqrt 10 - 4 sqrt 10*sqrt 2 = 3*3*2 *sqrt 5 + 3*3*2 - 4*10-4*2*sqrt 5 `

`= 18 sqrt 5 + 18 - 40 - 8 sqrt 5 = 10 sqrt 5 - 22`

b) `(2 sqrt 72 - 3 sqrt 18 + 5 sqrt 8)*3*sqrt 10 = (2*2*3*sqrt 2 - 3*3 sqrt 2 + 5*2 sqrt 2)*3*sqrt 10`

` = (12 sqrt 2 - 9 sqrt 2 + 10 sqrt 2)*3*sqrt 10 = (13 sqrt 2)*3*sqrt 10 = 39*2*sqrt 5 = 78 sqrt 5`

c) `(1 + sqrt 2 - sqrt 3)(1 - sqrt 2)-(1- sqrt 2 + sqrt 3)(1-sqrt 3)= (1 + sqrt 2 - sqrt 3 - sqrt 2 - sqrt 2*sqrt 2 + sqrt 3 * sqrt 2) - (1 - sqrt 2 + sqrt 3 - sqrt 3 + sqrt 2*sqrt 3 - sqrt 3*sqrt 3)`

` = (1 - sqrt 3 + 2 + sqrt 6) - (1 - sqrt 2 + sqrt 6 - 3) = 3 -sqrt 3 + sqrt 6 + 2 + sqrt 2 - sqrt 6 = 5 - sqrt 3 + sqrt 2`




10
a) `sqrt (4a) - 4 sqrt a + sqrt (ab^2) = 2 sqrt a - 4 sqrt a + b sqrt a = (b-2) sqrt a`

b) `sqrt (a/(3b)) : sqrt(b^3/(27a)) = sqrt ((a*27 a)/(3 b*b^3)) = (3 a)/b^2`

c) `sqrt b + b sqrt b + b^2 sqrt b = (1 + b+ b^2) sqrt b`




11
a) `1/sqrt 2 = sqrt 2/2 ` b) `sqrt 3 * sqrt 2 / 2 = 1/2 sqrt 6`

c) `1/3 sqrt 6 ` d) `sqrt 5 ` e) `1/2 sqrt 5`


15
a) `sqrt r^2 = | r |

b) `sqrt (1-r)^2 = |1-r|`

c) `9 |r|`


12
a) `(1-sqrt 3)(1 + sqrt 3) = 1 - sqrt 3 +sqrt 3 - sqrt 3 * sqrt 3` = 1 - 3 = -2`

b) `(sqrt 3 + sqrt 108)^2 = (sqrt 3)^2 + 2 sqrt 3 * sqrt 108 + (sqrt 108)^2 = 3 + sqrt (2 * 2 * 3 * 3 * 3 * 3) + 108 = 3 + 2*9 + 108 = 129`

c) `((5 - sqrt 2)(5 + sqrt 2))/((sqrt 2 - sqrt 18)^2) = (25 - 2)/(2 - 2 sqrt 36 + 18) = 23/2`




20
a) `1/(x sqrt x) = sqrt x / (sqrt x * sqrt x * x) = sqrt x / (x^2)`

b) `(sqrt x + sqrt y)/sqrt(xy) = (sqrt(xy)*sqrt(x+y))/(sqrt(xy)*sqrt(xy)) = sqrt(xy*(x+y))/xy`

c) `sqrt x/sqrt y - (x -xy)/sqrt(xy) = (sqrt x*sqrt x)/(sqrt x * sqrt y)- (x -xy)/sqrt(xy) = (x - x + xy)/sqrt (xy) = (xy)/sqrt (xy) = (sqrt(xy)*sqrt(xy))/sqrt (xy) = sqrt(xy)`
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