Solve 2+6/sqrt{6+3sqrt{2}}=frac{2sqrt{6}}{2R}

$$\frac { 2 + 6 } { \sqrt { 6 } + 3 \sqrt { 2 } } = \frac { 2 \sqrt { 6 } } { 2 R }$$

Solve for R

$$R = \frac{3 {(\sqrt{3} + 1)}}{4} \approx 2.049038106$$

Steps for Solving Linear Equation

Variable R cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 2R.

$$-\frac{1}{6}R\left(6^{\frac{1}{2}}-3\times 2^{\frac{1}{2}}\right)\left(2+6\right)=2\sqrt{6}$$

Use the distributive property to multiply -\frac{1}{6}R by 6^{\frac{1}{2}}-3\times 2^{\frac{1}{2}}.

$$\left(-\frac{1}{6}R\times 6^{\frac{1}{2}}+\frac{1}{2}R\times 2^{\frac{1}{2}}\right)\left(2+6\right)=2\sqrt{6}$$

Add 2 and 6 to get 8.

$$\left(-\frac{1}{6}R\times 6^{\frac{1}{2}}+\frac{1}{2}R\times 2^{\frac{1}{2}}\right)\times 8=2\sqrt{6}$$

Use the distributive property to multiply -\frac{1}{6}R\times 6^{\frac{1}{2}}+\frac{1}{2}R\times 2^{\frac{1}{2}} by 8.

$$-\frac{4}{3}R\times 6^{\frac{1}{2}}+4R\times 2^{\frac{1}{2}}=2\sqrt{6}$$

Reorder the terms.

$$4\sqrt{2}R-\frac{4}{3}\sqrt{6}R=2\sqrt{6}$$

Combine all terms containing R.

$$\left(4\sqrt{2}-\frac{4}{3}\sqrt{6}\right)R=2\sqrt{6}$$

The equation is in standard form.

$$\left(-\frac{4\sqrt{6}}{3}+4\sqrt{2}\right)R=2\sqrt{6}$$

Divide both sides by 4\sqrt{2}-\frac{4}{3}\sqrt{6}.

$$\frac{\left(-\frac{4\sqrt{6}}{3}+4\sqrt{2}\right)R}{-\frac{4\sqrt{6}}{3}+4\sqrt{2}}=\frac{2\sqrt{6}}{-\frac{4\sqrt{6}}{3}+4\sqrt{2}}$$

Dividing by 4\sqrt{2}-\frac{4}{3}\sqrt{6} undoes the multiplication by 4\sqrt{2}-\frac{4}{3}\sqrt{6}.

$$R=\frac{2\sqrt{6}}{-\frac{4\sqrt{6}}{3}+4\sqrt{2}}$$

Divide 2\sqrt{6} by 4\sqrt{2}-\frac{4}{3}\sqrt{6}.

$$R=\frac{3\sqrt{3}+3}{4}$$