Solve 80/frac{sqrt{6-sqrt{2}}{4}}=x/frac{1{2}}

$$\frac { 80 } { \frac { \sqrt { 6 } - \sqrt { 2 } } { 4 } } = \frac { x } { \frac { 1 } { 2 } }$$

Solve for x

$$x=40\left(\sqrt{2}+\sqrt{6}\right)\approx 154.548132206$$

Steps for Solving Linear Equation

Divide 80 by \frac{\sqrt{6}-\sqrt{2}}{4} by multiplying 80 by the reciprocal of \frac{\sqrt{6}-\sqrt{2}}{4}.

$$\frac{80\times 4}{\sqrt{6}-\sqrt{2}}=\frac{x}{\frac{1}{2}}$$

Multiply 80 and 4 to get 320.

$$\frac{320}{\sqrt{6}-\sqrt{2}}=\frac{x}{\frac{1}{2}}$$

Rationalize the denominator of \frac{320}{\sqrt{6}-\sqrt{2}} by multiplying numerator and denominator by \sqrt{6}+\sqrt{2}.

$$\frac{320\left(\sqrt{6}+\sqrt{2}\right)}{\left(\sqrt{6}-\sqrt{2}\right)\left(\sqrt{6}+\sqrt{2}\right)}=\frac{x}{\frac{1}{2}}$$

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}.

$$\frac{320\left(\sqrt{6}+\sqrt{2}\right)}{\left(\sqrt{6}\right)^{2}-\left(\sqrt{2}\right)^{2}}=\frac{x}{\frac{1}{2}}$$

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

$$\frac{320\left(\sqrt{6}+\sqrt{2}\right)}{6-2}=\frac{x}{\frac{1}{2}}$$

Subtract 2 from 6 to get 4.

$$\frac{320\left(\sqrt{6}+\sqrt{2}\right)}{4}=\frac{x}{\frac{1}{2}}$$

Divide 320\left(\sqrt{6}+\sqrt{2}\right) by 4 to get 80\left(\sqrt{6}+\sqrt{2}\right).

$$80\left(\sqrt{6}+\sqrt{2}\right)=\frac{x}{\frac{1}{2}}$$

Use the distributive property to multiply 80 by \sqrt{6}+\sqrt{2}.

$$80\sqrt{6}+80\sqrt{2}=\frac{x}{\frac{1}{2}}$$

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

$$\frac{x}{\frac{1}{2}}=80\sqrt{6}+80\sqrt{2}$$

The equation is in standard form.

$$2x=80\sqrt{2}+80\sqrt{6}$$

Divide both sides by 2.

$$\frac{2x}{2}=\frac{80\sqrt{2}+80\sqrt{6}}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x=\frac{80\sqrt{2}+80\sqrt{6}}{2}$$

Divide 80\sqrt{6}+80\sqrt{2} by 2.

$$x=40\sqrt{2}+40\sqrt{6}$$