Solve frac{2sqrt{2}}{5-2sqrt{6}}

$$\frac{ 2 \sqrt{ 2 } }{ 5-2 \sqrt{ 6 } }$$

Evaluate

$$8\sqrt{3}+10\sqrt{2}\approx 27.998542084$$

Solution Steps

Rationalize the denominator of \frac{2\sqrt{2}}{5-2\sqrt{6}} by multiplying numerator and denominator by 5+2\sqrt{6}.

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

Consider \left(5-2\sqrt{6}\right)\left(5+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}.

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

Calculate 5 to the power of 2 and get 25.

$$\frac{2\sqrt{2}\left(5+2\sqrt{6}\right)}{25-\left(-2\sqrt{6}\right)^{2}}$$

Expand \left(-2\sqrt{6}\right)^{2}.

$$\frac{2\sqrt{2}\left(5+2\sqrt{6}\right)}{25-\left(-2\right)^{2}\left(\sqrt{6}\right)^{2}}$$

Calculate -2 to the power of 2 and get 4.

$$\frac{2\sqrt{2}\left(5+2\sqrt{6}\right)}{25-4\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{2\sqrt{2}\left(5+2\sqrt{6}\right)}{25-4\times 6}$$

Multiply 4 and 6 to get 24.

$$\frac{2\sqrt{2}\left(5+2\sqrt{6}\right)}{25-24}$$

Subtract 24 from 25 to get 1.

$$\frac{2\sqrt{2}\left(5+2\sqrt{6}\right)}{1}$$

Anything divided by one gives itself.

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

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

$$10\sqrt{2}+4\sqrt{2}\sqrt{6}$$

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}.

$$10\sqrt{2}+4\sqrt{2}\sqrt{2}\sqrt{3}$$

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

$$10\sqrt{2}+4\times 2\sqrt{3}$$

Multiply 4 and 2 to get 8.

$$10\sqrt{2}+8\sqrt{3}$$