Question

       Освобождения от иррациональности в знаменателе дроби

1) ( √x+√y)/(√x-√y) 

2) (20-4√a)/(5√a-a)

3) (9√a+√b)/(9b+81√ab)

4) (x-a√x)/(√ax-a√a)

Answer 1

1) 

(sqrt{x}-sqrt{y})/(sqrt{x}+sqrt{y})=
(sqrt{x}-sqrt{y})(sqrt{x}-sqrt{y})/(sqrt{x}+sqrt{y})(sqrt{x}-sqrt{y})=
(x-2sqrt{xy}+y)/(x-y)

2) 

(20-4sqrt{a})/(5sqrt{a}-a)=4(5-sqrt{a})/sqrt{a}(5-sqrt{a})=4sqrt{a}/a

3)

(9sqrt{a}+sqrt{b})/(9b+81sqrt{ab})=(9sqrt{a}+sqrt{b})/(9sqrt{b}(sqrt{b}+9sqrt{a})=sqrt{b}/9b

4) 

(x-asqrt{x})/(sqrt{ax}-asqrt{a})=sqrt{x}(sqrt{x}-a)/sqrt{a}(sqrt{x}-a)=sqrt{ax}/a

Answer 2

1) (√x+√y)/(√x-√y)=

= (√x+√y)²/[(√x-√y)·( √x+√y)] =

= (√x+√y)²/[(√x)²-(√y)²] =

= (√x+√y)²/(x-y)

 

2) (20-4√a)/(5√a-a) =

= 4(5 - √a)/(5√а - √a·√a)

= 4(5 - √a)/[√a(5 -√a)] =

= 4/√a =

= 4√a/(√a·√a) =

= 4√a/a

 

3) (9√a+√b)/(9b+81√ab) =

= (9√a+√b)/[9√b(√b+9√a]) =

= 1/(9√b) =

= √b/(9b)

 

4) (x-a√x)/(√ax-a√a) =

= √x(√x - a)/[√a(√x - a)] =

= √x/√a =

= √(ax)/a