Matematika házi

1. Feladat
a.)
`(-3)^-2=3^-2=1/3^2=color(red)(1/9)`

b.)
`(1/3)^-2=3^2=color(red)(9)`

c.)
`(-2/3)^-1=color(red)(3/2)`

d.)
`(5^-3)^-2*(5^4)^-3*5^7=5^6*5^-12*5^7=color(red)(5)`



2. Feladat
a.)
`3sqrt(2/3)=sqrt(2/3*3^2)=sqrt(2/3*9)=color(red)(sqrt(6))`

b.)
`5sqrt(3/7)=sqrt(3/7*5^2)=sqrt(3/7*25)=color(red)(sqrt(75/7))`

c.)
`15sqrt(7/30)=sqrt(7/30*15^2)=sqrt(7/30*225)=sqrt(1575/30)=color(red)(sqrt(315/6))`

d.)
`2x^2sqrt(3xy)=sqrt(3xy*(2x^2)^2)=sqrt(3xy*4x^4)=color(red)(sqrt(12x^5y))`

e.)
`11xsqrt(2x^3y)=sqrt(2x^3y*(11x)^2)=sqrt(2x^3y*121x^2)=color(red)(sqrt(242x^5y))`



3. Feladat
a.)
`2/(2+sqrt2)=2/(2+sqrt2)*(2-sqrt2)/(2-sqrt2)=(2*(2-sqrt2))/((2+sqrt2)*(2-sqrt2))=(2*(2-sqrt2))/(4-2)=(2*(2-sqrt2))/2=color(red)(2-sqrt2)`

b.)
`9/(sqrt19-4)=9/(sqrt19-4)*(sqrt19+4)/(sqrt19+4)=(9*(sqrt19+4))/((sqrt19-4)*(sqrt19+4))=(9*(sqrt19+4))/(19-16)=(9*(sqrt19+4))/3=3*(sqrt19+4)=color(red)(3sqrt19+12)`

c.)
`6/(sqrt5+sqrt2)=6/(sqrt5+sqrt2)*(sqrt5-sqrt2)/(sqrt5-sqrt2)=(6*(sqrt5+sqrt2))/((sqrt5+sqrt2)*(sqrt5-sqrt2))=(6*(sqrt5+sqrt2))/(5-2)=(6*(sqrt5+sqrt2))/3=2*(sqrt5-sqrt2)=color(red)(2sqrt5-2sqrt2)`