Solve frac{(1+sqrt{12})^2-(2-sqrt{3})(sqrt{3}-2)}{(sqrt{8}-2)(2+2sqrt{2})}

$$\frac { ( 1 + \sqrt { 12 } ) ^ { 2 } - ( 2 - \sqrt { 3 } ) ( \sqrt { 3 } - 2 ) } { ( \sqrt { 8 } - 2 ) ( 2 + 2 \sqrt { 2 } ) }$$

Evaluate

$$5$$

Solution Steps

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

$$\frac{\left(1+2\sqrt{3}\right)^{2}-\left(2-\sqrt{3}\right)\left(\sqrt{3}-2\right)}{\left(\sqrt{8}-2\right)\left(2+2\sqrt{2}\right)}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+2\sqrt{3}\right)^{2}.

$$\frac{1+4\sqrt{3}+4\left(\sqrt{3}\right)^{2}-\left(2-\sqrt{3}\right)\left(\sqrt{3}-2\right)}{\left(\sqrt{8}-2\right)\left(2+2\sqrt{2}\right)}$$

The square of \sqrt{3} is 3.

$$\frac{1+4\sqrt{3}+4\times 3-\left(2-\sqrt{3}\right)\left(\sqrt{3}-2\right)}{\left(\sqrt{8}-2\right)\left(2+2\sqrt{2}\right)}$$

Multiply 4 and 3 to get 12.

$$\frac{1+4\sqrt{3}+12-\left(2-\sqrt{3}\right)\left(\sqrt{3}-2\right)}{\left(\sqrt{8}-2\right)\left(2+2\sqrt{2}\right)}$$

Add 1 and 12 to get 13.

$$\frac{13+4\sqrt{3}-\left(2-\sqrt{3}\right)\left(\sqrt{3}-2\right)}{\left(\sqrt{8}-2\right)\left(2+2\sqrt{2}\right)}$$

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}.

$$\frac{13+4\sqrt{3}-\left(2-\sqrt{3}\right)\left(\sqrt{3}-2\right)}{\left(2\sqrt{2}-2\right)\left(2+2\sqrt{2}\right)}$$

Use the distributive property to multiply 2-\sqrt{3} by \sqrt{3}-2 and combine like terms.

$$\frac{13+4\sqrt{3}-\left(4\sqrt{3}-4-\left(\sqrt{3}\right)^{2}\right)}{\left(2\sqrt{2}-2\right)\left(2+2\sqrt{2}\right)}$$

The square of \sqrt{3} is 3.

$$\frac{13+4\sqrt{3}-\left(4\sqrt{3}-4-3\right)}{\left(2\sqrt{2}-2\right)\left(2+2\sqrt{2}\right)}$$

Subtract 3 from -4 to get -7.

$$\frac{13+4\sqrt{3}-\left(4\sqrt{3}-7\right)}{\left(2\sqrt{2}-2\right)\left(2+2\sqrt{2}\right)}$$

To find the opposite of 4\sqrt{3}-7, find the opposite of each term.

$$\frac{13+4\sqrt{3}-4\sqrt{3}+7}{\left(2\sqrt{2}-2\right)\left(2+2\sqrt{2}\right)}$$

Combine 4\sqrt{3} and -4\sqrt{3} to get 0.

$$\frac{13+7}{\left(2\sqrt{2}-2\right)\left(2+2\sqrt{2}\right)}$$

Add 13 and 7 to get 20.

$$\frac{20}{\left(2\sqrt{2}-2\right)\left(2+2\sqrt{2}\right)}$$

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

$$\frac{20}{4\left(\sqrt{2}\right)^{2}-4}$$

The square of \sqrt{2} is 2.

$$\frac{20}{4\times 2-4}$$

Multiply 4 and 2 to get 8.

$$\frac{20}{8-4}$$

Subtract 4 from 8 to get 4.

$$\frac{20}{4}$$

Divide 20 by 4 to get 5.

$$5$$

Factor

$$5$$