Освобождения от иррациональности в знаменателе дроби 1) ( √x+√y)/(√x-√y) 2)...

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) (√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

Answer 2

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

2) 

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

3)

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

4) 

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