Question

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Answer 1

$\frac{a^2-ab}{a+b+2 \sqrt{ab} } : \frac{a^{- \frac{1}{2}} -b^{- \frac{1}{2} }}{a^{- \frac{1}{2}}+b^{- \frac{1}{2} }} =-a$

$1)$
$\frac{a^2-ab}{a+b+2 \sqrt{ab} } = \frac{a(a-b)}{( \sqrt{a})^2+ (\sqrt{b} )^2 +2 \sqrt{ab} } = \frac{a( \sqrt{a}- \sqrt{b})( \sqrt{a}+ \sqrt{b})}{( \sqrt{a} + \sqrt{b})^2 }= \frac{a( \sqrt{a}- \sqrt{b})}{\sqrt{a} + \sqrt{b} }$

$2)$

$\frac{\frac{1}{ \sqrt{a} }- \frac{1}{ \sqrt{b} } } { \frac{1}{ \sqrt{a} }+ \frac{1}{ \sqrt{b}} }= \frac{ \sqrt{b}- \sqrt{a} } { \sqrt{ab} } :\frac{ \sqrt{b}+ \sqrt{a} } { \sqrt{ab} }=\frac{ \sqrt{b}- \sqrt{a} } { \sqrt{ab} } *\frac{ \sqrt{ab} } { \sqrt{b}+ \sqrt{a} }= \frac{ \sqrt{b}- \sqrt{a}}{ \sqrt{b}+ \sqrt{a}}$

$3)$

$\frac{a( \sqrt{a}- \sqrt{b})}{\sqrt{a} + \sqrt{b} }:\frac{ \sqrt{b}- \sqrt{a}}{ \sqrt{b}+ \sqrt{a}}=\frac{a( \sqrt{a}- \sqrt{b})}{\sqrt{a} + \sqrt{b} }* \frac{ \sqrt{b}+ \sqrt{a}}{ \sqrt{b}- \sqrt{a}}=\frac{a( \sqrt{a}- \sqrt{b})}{1 }* \frac{1}{ -(\sqrt{a}- \sqrt{b})}=$$=\frac{a( \sqrt{a}- \sqrt{b})}{-( \sqrt{a}- \sqrt{b}) }=-a$