Solve frac{{left(sqrt{4}+sqrt{8}right)}^2-4sqrt{4*8}}{sqrt{4}-sqrt{8}}

$$\frac{ { \left( \sqrt{ 4 } + \sqrt{ 8 } \right) }^{ 2 } -4 \sqrt{ 4 \cdot 8 } }{ \sqrt{ 4 } - \sqrt{ 8 } }$$

Evaluate

$$2-2\sqrt{2}\approx -0.828427125$$

Solution Steps

Calculate the square root of 4 and get 2.

$$\frac{\left(2+\sqrt{8}\right)^{2}-4\sqrt{4\times 8}}{\sqrt{4}-\sqrt{8}}$$

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}.

$$\frac{\left(2+2\sqrt{2}\right)^{2}-4\sqrt{4\times 8}}{\sqrt{4}-\sqrt{8}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2+2\sqrt{2}\right)^{2}.

$$\frac{4+8\sqrt{2}+4\left(\sqrt{2}\right)^{2}-4\sqrt{4\times 8}}{\sqrt{4}-\sqrt{8}}$$

The square of \sqrt{2} is 2.

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

Multiply 4 and 2 to get 8.

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

Add 4 and 8 to get 12.

$$\frac{12+8\sqrt{2}-4\sqrt{4\times 8}}{\sqrt{4}-\sqrt{8}}$$

Multiply 4 and 8 to get 32.

$$\frac{12+8\sqrt{2}-4\sqrt{32}}{\sqrt{4}-\sqrt{8}}$$

Factor 32=4^{2}\times 2. Rewrite the square root of the product \sqrt{4^{2}\times 2} as the product of square roots \sqrt{4^{2}}\sqrt{2}. Take the square root of 4^{2}.

$$\frac{12+8\sqrt{2}-4\times 4\sqrt{2}}{\sqrt{4}-\sqrt{8}}$$

Multiply 4 and 4 to get 16.

$$\frac{12+8\sqrt{2}-16\sqrt{2}}{\sqrt{4}-\sqrt{8}}$$

Combine 8\sqrt{2} and -16\sqrt{2} to get -8\sqrt{2}.

$$\frac{12-8\sqrt{2}}{\sqrt{4}-\sqrt{8}}$$

Calculate the square root of 4 and get 2.

$$\frac{12-8\sqrt{2}}{2-\sqrt{8}}$$

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}.

$$\frac{12-8\sqrt{2}}{2-2\sqrt{2}}$$

Rationalize the denominator of \frac{12-8\sqrt{2}}{2-2\sqrt{2}} by multiplying numerator and denominator by 2+2\sqrt{2}.

$$\frac{\left(12-8\sqrt{2}\right)\left(2+2\sqrt{2}\right)}{\left(2-2\sqrt{2}\right)\left(2+2\sqrt{2}\right)}$$

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

$$\frac{\left(12-8\sqrt{2}\right)\left(2+2\sqrt{2}\right)}{2^{2}-\left(-2\sqrt{2}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\left(12-8\sqrt{2}\right)\left(2+2\sqrt{2}\right)}{4-\left(-2\sqrt{2}\right)^{2}}$$

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

$$\frac{\left(12-8\sqrt{2}\right)\left(2+2\sqrt{2}\right)}{4-\left(-2\right)^{2}\left(\sqrt{2}\right)^{2}}$$

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

$$\frac{\left(12-8\sqrt{2}\right)\left(2+2\sqrt{2}\right)}{4-4\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(12-8\sqrt{2}\right)\left(2+2\sqrt{2}\right)}{4-4\times 2}$$

Multiply 4 and 2 to get 8.

$$\frac{\left(12-8\sqrt{2}\right)\left(2+2\sqrt{2}\right)}{4-8}$$

Subtract 8 from 4 to get -4.

$$\frac{\left(12-8\sqrt{2}\right)\left(2+2\sqrt{2}\right)}{-4}$$

Use the distributive property to multiply 12-8\sqrt{2} by 2+2\sqrt{2} and combine like terms.

$$\frac{24+8\sqrt{2}-16\left(\sqrt{2}\right)^{2}}{-4}$$

The square of \sqrt{2} is 2.

$$\frac{24+8\sqrt{2}-16\times 2}{-4}$$

Multiply -16 and 2 to get -32.

$$\frac{24+8\sqrt{2}-32}{-4}$$

Subtract 32 from 24 to get -8.

$$\frac{-8+8\sqrt{2}}{-4}$$