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

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

Commented by 7770 last updated on 17/Jul/21

$$\mathbf{let}{\mathbf{x}}^{0.5}=\mathbf{a}{\mathbf{y}}^{0.5}=\mathbf{b}{\mathbf{x}}^{\frac{1}{6}}=\mathbf{m}{\mathbf{y}}^{\frac{1}{6}}=\mathbf{n}$$$$\frac{\mathbf{a}+\mathbf{b}}{\mathbf{m}+\mathbf{n}}-\frac{\mathbf{a}-\mathbf{b}}{\mathbf{m}-\mathbf{n}}\Rightarrow \frac{{\mathbf{a}}^2-{\mathbf{b}}^2}{{\mathbf{m}}^2-{\mathbf{n}}^2}$$$$\frac{\mathbf{x}-\mathbf{y}}{\sqrt[3]{\mathbf{x}}-\sqrt[3]{\mathbf{y}}}$$

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

$$\frac{\mathbf{a}+\mathbf{b}}{\mathbf{m}+\mathbf{n}}-\frac{\mathbf{a}-\mathbf{b}}{\mathbf{m}-\mathbf{n}}\Rightarrow \frac{{\mathbf{a}}^2-{\mathbf{b}}^2}{{\mathbf{m}}^2-{\mathbf{n}}^2}???$$$$\frac{\mathbf{a}+\mathbf{b}}{\mathbf{m}+\mathbf{n}}\times \frac{\mathbf{a}-\mathbf{b}}{\mathbf{m}-\mathbf{n}}=\frac{{\mathbf{a}}^2-{\mathbf{b}}^2}{{\mathbf{m}}^2-{\mathbf{n}}^2}$$

Answered by bobhans last updated on 17/Jul/21

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

Commented by mathdanisur last updated on 17/Jul/21

$$thankyouSer$$

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