Solve 8divfrac{sqrt{6}-sqrt{2}}{4}

$$8 \div \frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }$$

Evaluate

$$8\left(\sqrt{2}+\sqrt{6}\right)\approx 30.909626441$$

Solution Steps

Divide 8 by \frac{\sqrt{6}-\sqrt{2}}{4} by multiplying 8 by the reciprocal of \frac{\sqrt{6}-\sqrt{2}}{4}.

$$\frac{8\times 4}{\sqrt{6}-\sqrt{2}}$$

Multiply 8 and 4 to get 32.

$$\frac{32}{\sqrt{6}-\sqrt{2}}$$

Rationalize the denominator of \frac{32}{\sqrt{6}-\sqrt{2}} by multiplying numerator and denominator by \sqrt{6}+\sqrt{2}.

$$\frac{32\left(\sqrt{6}+\sqrt{2}\right)}{\left(\sqrt{6}-\sqrt{2}\right)\left(\sqrt{6}+\sqrt{2}\right)}$$

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}.

$$\frac{32\left(\sqrt{6}+\sqrt{2}\right)}{\left(\sqrt{6}\right)^{2}-\left(\sqrt{2}\right)^{2}}$$

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

$$\frac{32\left(\sqrt{6}+\sqrt{2}\right)}{6-2}$$

Subtract 2 from 6 to get 4.

$$\frac{32\left(\sqrt{6}+\sqrt{2}\right)}{4}$$

Divide 32\left(\sqrt{6}+\sqrt{2}\right) by 4 to get 8\left(\sqrt{6}+\sqrt{2}\right).

$$8\left(\sqrt{6}+\sqrt{2}\right)$$

Use the distributive property to multiply 8 by \sqrt{6}+\sqrt{2}.

$$8\sqrt{6}+8\sqrt{2}$$