Solve y^6=(91/8)^frac{1{8}}x+(65)^frac{1{4}}+2

$$y ^ { 6 } = ( \frac { 91 } { 8 } ) ^ { \frac { 1 } { 8 } } x + ( 65 ) ^ { \frac { 1 } { 4 } } + 2$$

Solve for x (complex solution)

$$x=-\frac{91^{\frac{7}{8}}\sqrt[8]{8}\left(-y^{6}+\sqrt[4]{65}+2\right)}{91}$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$\left(\frac{91}{8}\right)^{\frac{1}{8}}x+65^{\frac{1}{4}}+2=y^{6}$$

Subtract 65^{\frac{1}{4}} from both sides.

$$\left(\frac{91}{8}\right)^{\frac{1}{8}}x+2=y^{6}-65^{\frac{1}{4}}$$

Subtract 2 from both sides.

$$\left(\frac{91}{8}\right)^{\frac{1}{8}}x=y^{6}-65^{\frac{1}{4}}-2$$

Reorder the terms.

$$\sqrt[8]{\frac{91}{8}}x=y^{6}-\sqrt[4]{65}-2$$

Divide both sides by \sqrt[8]{\frac{91}{8}}.

$$\frac{\sqrt[8]{\frac{91}{8}}x}{\sqrt[8]{\frac{91}{8}}}=\frac{y^{6}-\sqrt[4]{65}-2}{\sqrt[8]{\frac{91}{8}}}$$

Dividing by \sqrt[8]{\frac{91}{8}} undoes the multiplication by \sqrt[8]{\frac{91}{8}}.

$$x=\frac{y^{6}-\sqrt[4]{65}-2}{\sqrt[8]{\frac{91}{8}}}$$

Divide y^{6}-\sqrt[4]{65}-2 by \sqrt[8]{\frac{91}{8}}.

$$x=\frac{\sqrt[8]{8}\left(y^{6}-\sqrt[4]{65}-2\right)}{\sqrt[8]{91}}$$

Solve for x

$$x=-\frac{2^{\frac{3}{8}}\times 91^{\frac{7}{8}}\left(-y^{6}+\sqrt[4]{65}+2\right)}{91}$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$\left(\frac{91}{8}\right)^{\frac{1}{8}}x+65^{\frac{1}{4}}+2=y^{6}$$

Subtract 65^{\frac{1}{4}} from both sides.

$$\left(\frac{91}{8}\right)^{\frac{1}{8}}x+2=y^{6}-65^{\frac{1}{4}}$$

Subtract 2 from both sides.

$$\left(\frac{91}{8}\right)^{\frac{1}{8}}x=y^{6}-65^{\frac{1}{4}}-2$$

Reorder the terms.

$$\sqrt[8]{\frac{91}{8}}x=y^{6}-\sqrt[4]{65}-2$$

Divide both sides by \sqrt[8]{\frac{91}{8}}.

$$\frac{\sqrt[8]{\frac{91}{8}}x}{\sqrt[8]{\frac{91}{8}}}=\frac{y^{6}-\sqrt[4]{65}-2}{\sqrt[8]{\frac{91}{8}}}$$

Dividing by \sqrt[8]{\frac{91}{8}} undoes the multiplication by \sqrt[8]{\frac{91}{8}}.

$$x=\frac{y^{6}-\sqrt[4]{65}-2}{\sqrt[8]{\frac{91}{8}}}$$

Divide y^{6}-\sqrt[4]{65}-2 by \sqrt[8]{\frac{91}{8}}.

$$x=\frac{\sqrt[8]{8}\left(y^{6}-\sqrt[4]{65}-2\right)}{\sqrt[8]{91}}$$

Solve for y (complex solution)

$$y\in \frac{2^{\frac{5}{6}}e^{\frac{\pi i}{3}}\sqrt[6]{2^{\frac{5}{8}}\sqrt[8]{91}x+2\sqrt[4]{65}+4}}{2},\frac{2^{\frac{5}{6}}\sqrt[6]{2^{\frac{5}{8}}\sqrt[8]{91}x+2\sqrt[4]{65}+4}}{2},\frac{2^{\frac{5}{6}}e^{\frac{2\pi i}{3}}\sqrt[6]{2^{\frac{5}{8}}\sqrt[8]{91}x+2\sqrt[4]{65}+4}}{2},-\frac{2^{\frac{5}{6}}\sqrt[6]{2^{\frac{5}{8}}\sqrt[8]{91}x+2\sqrt[4]{65}+4}}{2},\frac{2^{\frac{5}{6}}e^{\frac{4\pi i}{3}}\sqrt[6]{2^{\frac{5}{8}}\sqrt[8]{91}x+2\sqrt[4]{65}+4}}{2},\frac{2^{\frac{5}{6}}e^{\frac{5\pi i}{3}}\sqrt[6]{2^{\frac{5}{8}}\sqrt[8]{91}x+2\sqrt[4]{65}+4}}{2}$$

Solve for y

$$y=\frac{2^{\frac{5}{6}}\sqrt[6]{2^{\frac{5}{8}}\sqrt[8]{91}x+2\sqrt[4]{65}+4}}{2}$$

$$y=-\frac{2^{\frac{5}{6}}\sqrt[6]{2^{\frac{5}{8}}\sqrt[8]{91}x+2\sqrt[4]{65}+4}}{2}\text{, }x\geq -\frac{2^{\frac{3}{8}}\times 91^{\frac{7}{8}}\left(4\sqrt[4]{65}+8\right)}{364}$$