Exponentiële functies > Exponenten en machten
123456Exponenten en machten

Antwoorden van de opgaven

Opgave V1

`14`

Opgave 1
a

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

b

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

c

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

Opgave 2

`(31^25 * root[3](31^30))/(31^12)^3 = (31^25 * 31^(30/3))/(31^(12*3)) = (31^25 * 31^10)/(31^36) = (31^35)/(31^36) = 1/31`

Opgave 3
a

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

b

`8^(1 2/3) = 8^1 * 8^(2/3) = 8 * (root [3] (8))^2 = 8 * 2^2 = 8*4 = 32`

c

`4^(text(-)3)=1/(4^3)= 1/64`

d

`81^(1/4) = root [4] (81) = 3`

e

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

f

`(5^2*5^3)/(25^(1,5)) = (5^2*5^3)/(25^1*25^(1/2)) = (5^2*5^3)/(25*5) = (5^2*5^3)/(5^3) = 5^2 = 25 `

Opgave 4
a

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

b

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

c

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

d

`2 x^(2 1/2) = 2x^2 * x^(1/2) = 2x^2 * root[2](x^1) = 2x^2 sqrt(x)`

e

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

f

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

Opgave 5
a

`x^(text(-)3/4)`

b

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

c

`5x^(2/3)`

d

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

Opgave 6
a

In de exponent staat `2x + 1` . Voor exponentiële groei moet de formule de vorm `y = b*g^x` hebben.

b

`y=0,5^(2x+1) = 0,5^(2x) * 0,5^1 = (0,5^2)^x * 0,5 = 0,5*0,25^x`

Opgave 7
a

`y=2*3^(2x+2) = 2*(3^2)^x * 3^2 = 18*9^x`

b

`y=5*0,2^(3x-1) = 5*(0,2^3)^x * 0,2^(text(-)1) = 25*0,008^x`

c

`y=0,4*5^(text(-2)x+3) = 0,4*(5^(text(-)2))^x * 5^3 = 50*(1/25)^x`

Opgave 8
a

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

b

`2^3 * 2^2 = 2^(3+2) = 2^5 = 32`

c

`(2^(1/4))^8 = 2^(1/4*8) = 2^2 = 4`

d

`root[3](1000) = 1000^(1/3) = (10^3)^(1/3) = 10^(3*1/3) = 10^1 = 10`

Opgave 9
a

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

b

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

c

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

d

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

Opgave 10
a

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

b

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

c

`root [4] ( x^3 )`

d

`x root[4]( x^3 )`

e

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

f

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

Opgave 11
a

`(17^105)/(17^23) * 17^(text(-)85) = 17^82 * 17^(text(-)85) = 17^(82-85) = 17^(text(-)3) = 1/(17^3)`

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^(text(-)231)*4^230*9^(text(-)230)*3^233 = 4^(text(-)1)*3^464*3^(text(-)460) = (3^4)/4=81/4=` `20,25`

d

`(7^102)/((49^10)^5) = (7^102)/(((7^2)^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)^(1/2)*(root[3](4^3))^(1/2) = (4/9)^(1/2) * 4^(1/2) = sqrt(4/9) * 2 = 2/3 * 2 = 4/3`

f

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

Opgave 12
a

`y = (2 x^3)^4 * text(-)3 x^5 = 2^4*x^(3*4) * text(-)3x^5 = 16x^12 * text(-)3x^5 = 16*text(-)3*x^(12+5) = text(-)48x^17`

b

`y = (2 x * x^2)/(x^6) = (2x^(1+2))/(x^6) = (2x^3)/(x^6) = 2x^(3-6)= 2x^(text(-)3)`

c

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

d

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

Opgave 13
a

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

b

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

c

`y = (1/3)^(3 - 2 x) = (1/3)^3 * (1/3)^(text(-)2x) = 1/27 *((1/3)^(text(-)2))^x = 1/27 * 9^x`

d

`y = 6 * 2^(4 x - 2) = 6 * 2^(text(-)2) * 2^(4x) = 1,5 *(2^4)^x=1,5*16^x`

Opgave 14Exponentiële groei of niet?
Exponentiële groei of niet?
a

In de exponent staat `2x + 1` . Voor exponentiële groei moet de formule de vorm `y = b*g^x` hebben.

b

`y=4^(text(-)2x+1) = (4^(text(-)2))^x * 4^1 = 4*0,0625^x`

c

`y = 4^(text(-)2x)+1 = (4^(text(-)2))^x + 1 = 0,0625^x + 1`

Deze formule heeft niet de standaardvorm `y = b*g^x` .

Opgave 15
a

`5^(text(-)1)`

b

`7^5`

c

`5^4`

d

`5^(text(-)1)`

Opgave 16
a

`18 x^10`

b

`5 x^(text(-)1)`

c

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

d

`5 x^( 1 2/5)`

Opgave 17
a

`x*root(4)(x)`

b

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

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