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

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

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

`(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`

`(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`

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

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

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

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

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

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

`2 x^ (2 1/3)=2x^2*x^(1/3) = 2 x^2*root3 (x)` of `2*x^(7/3) = 2 root[3](x^7)` .

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

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

`2 sqrt(x)`

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

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

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

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

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

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

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

`2^5 = 32`

`2^2 = 4`

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

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

`root[4](x^3)`

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

`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^2 root[4](x^3))`

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

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

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

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

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

`(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`

`(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`

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

`(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`

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

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

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

`(3x^2*root[3](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)`

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

`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)`

`(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`

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

`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`

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

`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` .

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

• `(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`

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

`2^5=32`

`2^4=16`

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

`1/sqrt(x)`

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

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

`0,25 x^(1,5)`

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