Simplify-x-1-2-y-1-2-x-1-6-y-1-6-x-1-2-y-1-2-x-1-6-y-1-6-

Question Number 146989 by mathdanisur last updated on 16/Jul/21

$${Simplify}: \\ $$$$\frac{{x}^{\frac{\mathrm{1}}{\mathrm{2}}} \:+\:{y}^{\frac{\mathrm{1}}{\mathrm{2}}} }{{x}^{\frac{\mathrm{1}}{\mathrm{6}}} \:+\:{y}^{\frac{\mathrm{1}}{\mathrm{6}}} }\:\:-\:\:\frac{{x}^{\frac{\mathrm{1}}{\mathrm{2}}} \:-\:{y}^{\frac{\mathrm{1}}{\mathrm{2}}} }{{x}^{\frac{\mathrm{1}}{\mathrm{6}}} \:-\:{y}^{\frac{\mathrm{1}}{\mathrm{6}}} }\:=\:? \\ $$

Commented by 7770 last updated on 17/Jul/21

let x^(0.5) =a y^(0.5) =b x^(1/6) =m y^(1/6) =n ((a+b)/(m+n))−((a−b)/(m−n))⇒((a^2 −b^2 )/(m^2 −n^2 )) ((x−y)/( (x)^(1/3) −(y)^(1/3) ))

$$\boldsymbol{\mathrm{let}}\:\boldsymbol{\mathrm{x}}^{\mathrm{0}.\mathrm{5}} =\boldsymbol{\mathrm{a}}\:\:\:\:\boldsymbol{\mathrm{y}}^{\mathrm{0}.\mathrm{5}} =\boldsymbol{\mathrm{b}}\:\:\:\:\boldsymbol{\mathrm{x}}^{\frac{\mathrm{1}}{\mathrm{6}}} =\boldsymbol{\mathrm{m}}\:\:\boldsymbol{\mathrm{y}}^{\frac{\mathrm{1}}{\mathrm{6}}} =\boldsymbol{\mathrm{n}} \\ $$$$\frac{\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{b}}}{\boldsymbol{\mathrm{m}}+\boldsymbol{\mathrm{n}}}−\frac{\boldsymbol{\mathrm{a}}−\boldsymbol{\mathrm{b}}}{\boldsymbol{\mathrm{m}}−\boldsymbol{\mathrm{n}}}\Rightarrow\frac{\boldsymbol{\mathrm{a}}^{\mathrm{2}} −\boldsymbol{\mathrm{b}}^{\mathrm{2}} }{\boldsymbol{\mathrm{m}}^{\mathrm{2}} −\boldsymbol{\mathrm{n}}^{\mathrm{2}} } \\ $$$$\frac{\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{y}}}{\:\sqrt[{\mathrm{3}}]{\boldsymbol{\mathrm{x}}}−\sqrt[{\mathrm{3}}]{\boldsymbol{\mathrm{y}}}} \\ $$

Commented by Rasheed.Sindhi last updated on 17/Jul/21

((a+b)/(m+n))−((a−b)/(m−n))⇒((a^2 −b^2 )/(m^2 −n^2 )) ??? ((a+b)/(m+n))×((a−b)/(m−n))=((a^2 −b^2 )/(m^2 −n^2 ))

$$\frac{\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{b}}}{\boldsymbol{\mathrm{m}}+\boldsymbol{\mathrm{n}}}−\frac{\boldsymbol{\mathrm{a}}−\boldsymbol{\mathrm{b}}}{\boldsymbol{\mathrm{m}}−\boldsymbol{\mathrm{n}}}\Rightarrow\frac{\boldsymbol{\mathrm{a}}^{\mathrm{2}} −\boldsymbol{\mathrm{b}}^{\mathrm{2}} }{\boldsymbol{\mathrm{m}}^{\mathrm{2}} −\boldsymbol{\mathrm{n}}^{\mathrm{2}} }\:??? \\ $$$$\frac{\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{b}}}{\boldsymbol{\mathrm{m}}+\boldsymbol{\mathrm{n}}}×\frac{\boldsymbol{\mathrm{a}}−\boldsymbol{\mathrm{b}}}{\boldsymbol{\mathrm{m}}−\boldsymbol{\mathrm{n}}}=\frac{\boldsymbol{\mathrm{a}}^{\mathrm{2}} −\boldsymbol{\mathrm{b}}^{\mathrm{2}} }{\boldsymbol{\mathrm{m}}^{\mathrm{2}} −\boldsymbol{\mathrm{n}}^{\mathrm{2}} } \\ $$

Answered by bobhans last updated on 17/Jul/21

{ ((x^(1/6) =u)),((y^(1/6) =v)) :} ⇒ ((u^3 +v^3 )/(u+v)) − ((u^3 −v^3 )/(u−v))=u^2 −uv+v^2 −(u^2 +uv+v^2 ) = −2uv =−2(xy)^(1/6)

$$\begin{cases}{\mathrm{x}^{\mathrm{1}/\mathrm{6}} =\mathrm{u}}\\{\mathrm{y}^{\mathrm{1}/\mathrm{6}} =\mathrm{v}}\end{cases} \\ $$$$\Rightarrow\:\frac{\mathrm{u}^{\mathrm{3}} +\mathrm{v}^{\mathrm{3}} }{\mathrm{u}+\mathrm{v}}\:−\:\frac{\mathrm{u}^{\mathrm{3}} −\mathrm{v}^{\mathrm{3}} }{\mathrm{u}−\mathrm{v}}=\mathrm{u}^{\mathrm{2}} −\mathrm{uv}+\mathrm{v}^{\mathrm{2}} −\left(\mathrm{u}^{\mathrm{2}} +\mathrm{uv}+\mathrm{v}^{\mathrm{2}} \right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:−\mathrm{2uv}\:=−\mathrm{2}\left(\mathrm{xy}\right)^{\mathrm{1}/\mathrm{6}} \\ $$

Commented by mathdanisur last updated on 17/Jul/21

$${thank}\:{you}\:{Ser} \\ $$

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