Solve a/sqrt{2/2}=frac{sqrt{8}sqrt{3}}{frac{sqrt{6}+sqrt{2}}{4}}

$$\frac { a } { \frac { \sqrt { 2 } } { 2 } } = \frac { \sqrt { 8 } \sqrt { 3 } } { \frac { \sqrt { 6 } + \sqrt { 2 } } { 4 } }$$

Solve for a

$$a=6\sqrt{2}-2\sqrt{6}\approx 3.586301889$$

Steps for Solving Linear Equation

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

$$\frac{a\times 2}{\sqrt{2}}=\frac{\sqrt{8}\sqrt{3}}{\frac{\sqrt{6}+\sqrt{2}}{4}}$$

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

$$\frac{a\times 2\sqrt{2}}{\left(\sqrt{2}\right)^{2}}=\frac{\sqrt{8}\sqrt{3}}{\frac{\sqrt{6}+\sqrt{2}}{4}}$$

The square of \sqrt{2} is 2.

$$\frac{a\times 2\sqrt{2}}{2}=\frac{\sqrt{8}\sqrt{3}}{\frac{\sqrt{6}+\sqrt{2}}{4}}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$\frac{a\times 2\sqrt{2}}{2}=\frac{2\sqrt{2}\sqrt{3}}{\frac{\sqrt{6}+\sqrt{2}}{4}}$$

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{a\times 2\sqrt{2}}{2}=\frac{2\sqrt{6}}{\frac{\sqrt{6}+\sqrt{2}}{4}}$$

Divide 2\sqrt{6} by \frac{\sqrt{6}+\sqrt{2}}{4} by multiplying 2\sqrt{6} by the reciprocal of \frac{\sqrt{6}+\sqrt{2}}{4}.

$$\frac{a\times 2\sqrt{2}}{2}=\frac{2\sqrt{6}\times 4}{\sqrt{6}+\sqrt{2}}$$

Rationalize the denominator of \frac{2\sqrt{6}\times 4}{\sqrt{6}+\sqrt{2}} by multiplying numerator and denominator by \sqrt{6}-\sqrt{2}.

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

Consider \left(\sqrt{6}+\sqrt{2}\right)\left(\sqrt{6}-\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{a\times 2\sqrt{2}}{2}=\frac{2\sqrt{6}\times 4\left(\sqrt{6}-\sqrt{2}\right)}{\left(\sqrt{6}\right)^{2}-\left(\sqrt{2}\right)^{2}}$$

Square \sqrt{6}. Square \sqrt{2}.

$$\frac{a\times 2\sqrt{2}}{2}=\frac{2\sqrt{6}\times 4\left(\sqrt{6}-\sqrt{2}\right)}{6-2}$$

Subtract 2 from 6 to get 4.

$$\frac{a\times 2\sqrt{2}}{2}=\frac{2\sqrt{6}\times 4\left(\sqrt{6}-\sqrt{2}\right)}{4}$$

Multiply 2 and 4 to get 8.

$$\frac{a\times 2\sqrt{2}}{2}=\frac{8\sqrt{6}\left(\sqrt{6}-\sqrt{2}\right)}{4}$$

Divide 8\sqrt{6}\left(\sqrt{6}-\sqrt{2}\right) by 4 to get 2\sqrt{6}\left(\sqrt{6}-\sqrt{2}\right).

$$\frac{a\times 2\sqrt{2}}{2}=2\sqrt{6}\left(\sqrt{6}-\sqrt{2}\right)$$

Use the distributive property to multiply 2\sqrt{6} by \sqrt{6}-\sqrt{2}.

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

The square of \sqrt{6} is 6.

$$\frac{a\times 2\sqrt{2}}{2}=2\times 6-2\sqrt{6}\sqrt{2}$$

Multiply 2 and 6 to get 12.

$$\frac{a\times 2\sqrt{2}}{2}=12-2\sqrt{6}\sqrt{2}$$

Factor 6=2\times 3. Rewrite the square root of the product \sqrt{2\times 3} as the product of square roots \sqrt{2}\sqrt{3}.

$$\frac{a\times 2\sqrt{2}}{2}=12-2\sqrt{2}\sqrt{3}\sqrt{2}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\frac{a\times 2\sqrt{2}}{2}=12-2\times 2\sqrt{3}$$

Multiply -2 and 2 to get -4.

$$\frac{a\times 2\sqrt{2}}{2}=12-4\sqrt{3}$$

Cancel out 2 and 2.

$$a\sqrt{2}=12-4\sqrt{3}$$

The equation is in standard form.

$$\sqrt{2}a=12-4\sqrt{3}$$

Divide both sides by \sqrt{2}.

$$\frac{\sqrt{2}a}{\sqrt{2}}=\frac{12-4\sqrt{3}}{\sqrt{2}}$$

Dividing by \sqrt{2} undoes the multiplication by \sqrt{2}.

$$a=\frac{12-4\sqrt{3}}{\sqrt{2}}$$

Divide 12-4\sqrt{3} by \sqrt{2}.

$$a=6\sqrt{2}-2\sqrt{6}$$