Exponentiële functies > Exponenten en machten
123456Exponenten en machten

Antwoorden van de opgaven

Opgave V1

`19`

Opgave 1
a

`root[n](g)=g^ (1/n)`

b

`(g^a) ^b=g^ (ab)`

c

`g^a/(g^b)=g^ (a-b)`

Opgave 2
a

`(31^25*root[3] (31^30)) /((31^12) ^3)=(31^25*(31^30)^(1/3))/(31^(12*3))=(31^(25+10))/(31^36)=1/31`

b

`(25^4*(5^5)^9)/(root [4](5^40)*125^14)=((5^2)^4*5^45)/((5^(40))^(1/4)*(5^3)^(14))=(5^8*5^45)/(5^10*5^42)=(5^53)/(5^52)=5`

Opgave 3
a

`3^(text(-)2)=1/(3^2)=1/9`

b

`81^(1/4)=(3^4)^(1/4)=3`

c

`8^(1 2/3)=8^(5/3)=8*8^(2/3)=8*(2^3)^(2/3)=32`

d

`2^(text(-)3)*2^7=2^(text(-)3+7)=2^4=16`

e

`(3^(text(-)12))^(1/4)=3^(text(-)3)=1/27`

f

`(2^ (1/2)) ^10=2^5=32`

Opgave 4
a

`2 x^ (2 1/3)=2x^2*x^(1/3)=2 x^2*root3 (x)`

b

`(3 x^(text(-)1)) / (2 x)=3*1/x * 1/(2x)=3/(2x^2)`

c

`4 x^(text(-)3/4)=4*1/(x^(3/4))` = `4/(root[4] (x^3))`

d

`2 sqrt(x)`

Opgave 5
a

`3/(2x)=3/2 * 1/x = 1,5x^(text(-)1)`

b

`3/ (2 xsqrt(x))=3/(2x^(1 1/2))=1,5x^(text(-)1 1/2)`

c

`(4 root[3] (x)) ^2=(4*x^(1/3))^2=4^2*(x^(1/3))^2=16x^(2/3)`

d

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

e

`2/ (x^3*root[3] (x^2))=2*x^(text(-)3)*x^(text(-)2/3)=2 x^ (text(-)3 2/3)`

f

`3 x^5* (2 x^3) ^2=3x^5*2^2*x^6=12x^11`

Opgave 6

`12 *3^ (text(-)0,5 x+1) =12 *3^ (text(-)0,5 x) *3^1=12 *3 * (3^(text(-)0,5)) ^x=36 * (1/ sqrt(3 ) ) ^x`

`b=36` en `g=1/ sqrt(3 )` .

Opgave 7
a

`(2^3) ^2=2^6=64`

b

`2^5=32`

c

`2^2=4`

d

`(10^3) ^ (1/3) =10^1=10`

Opgave 8
a

`x^(text(-)1/2)=1/x^(1/2)=1/sqrt(x)`

b

`root4 (x^3)`

c

`3x^(text(-)1,5)=3*1/(x^1,5)=3/(xsqrt(x))`

d

`1/2x^(text(-)2,75)=1/2*1/(x^(2,75))=1/2*1/(x^(2 3/4))=1/2*1/(x^2*root[4](x^3))=1/ (2 x^2root[4] (x^3))`

Opgave 9
a

`1/ (x^2sqrt(x))=1/x^(2,5)=x^(text(-)2,5)`

b

`1/ (3 *root4 (x))=1/3*1/root4(x)=1/3*1/x^(1/4)=1/3*x^(text(-)1/4)`

c

`1/ (2 xsqrt(x))=1/(2x^(1,5))=1/2*x^(text(-)1 1/2)`

d

`(3 xsqrt(x)) ^3=(3*x^(1,5))^3=3^3*x^(4 1/2)=27x^(4 1/2)`

Opgave 10
a

`(17^105)/(17^22)*17^(text(-)85)=17^83*17^(text(-)85)=17^(text(-)2)=1/17^2=1/289`

b

`(1/2) ^219*8^72=(2^(text(-)1))^219*(2^3)^72=2^(text(-)219)*2^216=2^(text(-)3)=1/2^3=1/8`

c

`(3/4)^231* (4/9)^230*3^233=(3^231)/(4^231)*(4^230)/(9^230)*3^233=(4^230)/(4^231)*(3^231)/(3^460)*3^233=1/4* 3^4=20 1/4`

d

`(7^102)/((49^10) ^5)=(7^102)/(((7^2)^10)^5)=(7^102)/(7^100)=7^2=49`

e

`(4/9*root[3](64))^(1/2)=(4/(9)*root[3](4^3))^(1/2)=(4/9*4)^(1/2)=(16/9)^(1/2)=4/3`

f

`(5^3*(3^5)^15)/(25*root(3)(3^225))= (5^3*3^75)/(5^2*3^75) =5`

Opgave 11
a

`(2 x^3) ^4*text(-)3 x^5=2^4*x^12*text(-)3x^5=text(-)48 x^17`

b

`(2 x*x^2) /x^4=(2x^3)/x^3=2 x^(text(-)1)`

c

`(3x^2*root3 (x))/(9x^3*x^(2/3)) =(3*x^2*x^(1/3))/(3^2*x^3*x^(2/3)) =(3*x^(7/3)) / (3^2*x^(11/3))=1/3 x^(text(-)4/3)`

d

`(2x^(1 1/2))^3/ (xsqrt(x)) =(8*x^(4 1/2))/ (x^(1 1/2))= 8x^3`

e

`1/2x^ (text(-)1 1/3) *root(4)(x) = 1/2 * x^(text(-)4/3) * x^(1/4) = 1/2 * x^(text(-)16/12) * x^(3/12)= 1/2x^(text(-)13/12)`

f

`(12x^(1 2/5))/(6x^(text(-)3/5))+2x^2= (12 x ^(7/5))/ (6x^(text(-)3/5)) + 2x^2 = 2x^2+2x^2=4x^2`

Opgave 12
a

`f(x)=3*2^(0,5x)=3*(2^(0,5))^x=3*(sqrt(2))^x`

b

`f(x)=0,5^(text(-)x+2)=0,5^(text(-)1*x)*0,5^2=(0,5^(text(-)1))^x* 0,25=0,25*2^x`

c

`f(x)=9*(1/3)^(4-2x)=3^2*(3^(text(-)1))^4*((3^(text(-)1))^(text(-)2))^x=1/9*9^x`

d

`f(x)=7*5^(2x+1)+2*(sqrt(5))^(4x)=7*(5^2)^x*5+2*((sqrt(5))^4)^x=35*25^x+2*25^x=37*25^x` .

Opgave 13Priemfactoren
Priemfactoren
a

`26=2*13`
`25=5^2`
`144=2^4*3^2`
`127=127`
`202=2*101`

b
  • `((126)^2*(3773))/(sqrt(81)*343*693)=((2*3^2*7)^2*7^3*11)/(3^2*7^3*3^2*7*11)=(2^2*3^4*7^5*11)/(3^4*7^4*11)=28`

  • `((5)^(text(-)2)*(19600)^(1/2))/(91)=((5)^(text(-)2)*(2^4*5^2*7^2)^(1/2))/(7*13)=(2^2*5^(text(-)1)*7)/(7*13)=4/(5*13)=4/65`

  • `((1080)^(1/3)*sqrt(5^(4/3)))/((1296)^(1/4))=((2^3*3^3*5)^(1/3)*(5^(4/3))^(1/2))/(2^4*3^4)^(1/4)=(2*3*5^(1/3)*5^(2/3))/(2*3)=5^(1/3+2/3)=5`

Opgave 14
a

`3^(text(-)1)=1/3`

b

`2^5=32`

c

`2^4=16`

d

`3^(text(-)1)=1/3`

Opgave 15
a

`1/sqrt(x)`

b

`x*root(10)(x^3)`

Opgave 16
a

`0,3 x^(text(-)1)`

b

`1/4x^(1,5)`

c

`1,5 x^ (text(-)1 1/2)`

Opgave 17

`f(x)=12sqrt(3) *3^x`

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