Solve 10/sqrt{8-sqrt{6}}

$$\frac { 10 } { \sqrt { 8 } - \sqrt { 6 } }$$

Evaluate

$$5\left(\sqrt{6}+2\sqrt{2}\right)\approx 26.389584338$$

Solution Steps

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}.

$$\frac{10}{2\sqrt{2}-\sqrt{6}}$$

Rationalize the denominator of \frac{10}{2\sqrt{2}-\sqrt{6}} by multiplying numerator and denominator by 2\sqrt{2}+\sqrt{6}.

$$\frac{10\left(2\sqrt{2}+\sqrt{6}\right)}{\left(2\sqrt{2}-\sqrt{6}\right)\left(2\sqrt{2}+\sqrt{6}\right)}$$

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

$$\frac{10\left(2\sqrt{2}+\sqrt{6}\right)}{\left(2\sqrt{2}\right)^{2}-\left(\sqrt{6}\right)^{2}}$$

Expand \left(2\sqrt{2}\right)^{2}.

$$\frac{10\left(2\sqrt{2}+\sqrt{6}\right)}{2^{2}\left(\sqrt{2}\right)^{2}-\left(\sqrt{6}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{10\left(2\sqrt{2}+\sqrt{6}\right)}{4\left(\sqrt{2}\right)^{2}-\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{10\left(2\sqrt{2}+\sqrt{6}\right)}{4\times 2-\left(\sqrt{6}\right)^{2}}$$

Multiply 4 and 2 to get 8.

$$\frac{10\left(2\sqrt{2}+\sqrt{6}\right)}{8-\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{10\left(2\sqrt{2}+\sqrt{6}\right)}{8-6}$$

Subtract 6 from 8 to get 2.

$$\frac{10\left(2\sqrt{2}+\sqrt{6}\right)}{2}$$

Divide 10\left(2\sqrt{2}+\sqrt{6}\right) by 2 to get 5\left(2\sqrt{2}+\sqrt{6}\right).

$$5\left(2\sqrt{2}+\sqrt{6}\right)$$

Use the distributive property to multiply 5 by 2\sqrt{2}+\sqrt{6}.

$$10\sqrt{2}+5\sqrt{6}$$