Solve frac{sqrt{2}}{frac{sqrt{5}}{3}}=x/sqrt{5/sqrt{6}}

$$\frac { \sqrt { 2 } } { \frac { \sqrt { 5 } } { 3 } } = \frac { x } { \frac { \sqrt { 5 } } { \sqrt { 6 } } }$$

Solve for x

$$x=\sqrt{3}\approx 1.732050808$$

Steps for Solving Linear Equation

Divide \sqrt{2} by \frac{\sqrt{5}}{3} by multiplying \sqrt{2} by the reciprocal of \frac{\sqrt{5}}{3}.

$$\frac{\sqrt{2}\times 3}{\sqrt{5}}=\frac{x}{\frac{\sqrt{5}}{\sqrt{6}}}$$

Rationalize the denominator of \frac{\sqrt{2}\times 3}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\sqrt{2}\times 3\sqrt{5}}{\left(\sqrt{5}\right)^{2}}=\frac{x}{\frac{\sqrt{5}}{\sqrt{6}}}$$

The square of \sqrt{5} is 5.

$$\frac{\sqrt{2}\times 3\sqrt{5}}{5}=\frac{x}{\frac{\sqrt{5}}{\sqrt{6}}}$$

To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{\sqrt{10}\times 3}{5}=\frac{x}{\frac{\sqrt{5}}{\sqrt{6}}}$$

Divide x by \frac{\sqrt{5}}{\sqrt{6}} by multiplying x by the reciprocal of \frac{\sqrt{5}}{\sqrt{6}}.

$$\frac{\sqrt{10}\times 3}{5}=\frac{x\sqrt{6}}{\sqrt{5}}$$

Rationalize the denominator of \frac{x\sqrt{6}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\sqrt{10}\times 3}{5}=\frac{x\sqrt{6}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\sqrt{10}\times 3}{5}=\frac{x\sqrt{6}\sqrt{5}}{5}$$

To multiply \sqrt{6} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{\sqrt{10}\times 3}{5}=\frac{x\sqrt{30}}{5}$$

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

$$\frac{x\sqrt{30}}{5}=\frac{\sqrt{10}\times 3}{5}$$

Multiply both sides of the equation by 5.

$$x\sqrt{30}=\sqrt{10}\times 3$$

The equation is in standard form.

$$\sqrt{30}x=3\sqrt{10}$$

Divide both sides by \sqrt{30}.

$$\frac{\sqrt{30}x}{\sqrt{30}}=\frac{3\sqrt{10}}{\sqrt{30}}$$

Dividing by \sqrt{30} undoes the multiplication by \sqrt{30}.

$$x=\frac{3\sqrt{10}}{\sqrt{30}}$$

Divide 3\sqrt{10} by \sqrt{30}.

$$x=\sqrt{3}$$