Solve (2sqrt{3}-sqrt{2})^2+2sqrt{6}(sqrt{3}+2)-(sqrt{5}+1)(1-sqrt{7})

$$( 2 \sqrt { 3 } - \sqrt { 2 } ) ^ { 2 } + 2 \sqrt { 6 } ( \sqrt { 3 } + 2 ) - ( \sqrt { 5 } + 1 ) ( 1 - \sqrt { 7 } )$$

Evaluate

$$\sqrt{7}+\sqrt{35}+6\sqrt{2}+13-\sqrt{5}\approx 27.811044491$$

Solution Steps

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

$$4\left(\sqrt{3}\right)^{2}-4\sqrt{3}\sqrt{2}+\left(\sqrt{2}\right)^{2}+2\sqrt{6}\left(\sqrt{3}+2\right)-\left(\sqrt{5}+1\right)\left(1-\sqrt{7}\right)$$

The square of \sqrt{3} is 3.

$$4\times 3-4\sqrt{3}\sqrt{2}+\left(\sqrt{2}\right)^{2}+2\sqrt{6}\left(\sqrt{3}+2\right)-\left(\sqrt{5}+1\right)\left(1-\sqrt{7}\right)$$

Multiply 4 and 3 to get 12.

$$12-4\sqrt{3}\sqrt{2}+\left(\sqrt{2}\right)^{2}+2\sqrt{6}\left(\sqrt{3}+2\right)-\left(\sqrt{5}+1\right)\left(1-\sqrt{7}\right)$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$12-4\sqrt{6}+\left(\sqrt{2}\right)^{2}+2\sqrt{6}\left(\sqrt{3}+2\right)-\left(\sqrt{5}+1\right)\left(1-\sqrt{7}\right)$$

The square of \sqrt{2} is 2.

$$12-4\sqrt{6}+2+2\sqrt{6}\left(\sqrt{3}+2\right)-\left(\sqrt{5}+1\right)\left(1-\sqrt{7}\right)$$

Add 12 and 2 to get 14.

$$14-4\sqrt{6}+2\sqrt{6}\left(\sqrt{3}+2\right)-\left(\sqrt{5}+1\right)\left(1-\sqrt{7}\right)$$

Use the distributive property to multiply 2\sqrt{6} by \sqrt{3}+2.

$$14-4\sqrt{6}+2\sqrt{6}\sqrt{3}+4\sqrt{6}-\left(\sqrt{5}+1\right)\left(1-\sqrt{7}\right)$$

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

$$14-4\sqrt{6}+2\sqrt{3}\sqrt{2}\sqrt{3}+4\sqrt{6}-\left(\sqrt{5}+1\right)\left(1-\sqrt{7}\right)$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$14-4\sqrt{6}+2\times 3\sqrt{2}+4\sqrt{6}-\left(\sqrt{5}+1\right)\left(1-\sqrt{7}\right)$$

Multiply 2 and 3 to get 6.

$$14-4\sqrt{6}+6\sqrt{2}+4\sqrt{6}-\left(\sqrt{5}+1\right)\left(1-\sqrt{7}\right)$$

Combine -4\sqrt{6} and 4\sqrt{6} to get 0.

$$14+6\sqrt{2}-\left(\sqrt{5}+1\right)\left(1-\sqrt{7}\right)$$

Use the distributive property to multiply \sqrt{5}+1 by 1-\sqrt{7}.

$$14+6\sqrt{2}-\left(\sqrt{5}-\sqrt{5}\sqrt{7}+1-\sqrt{7}\right)$$

To multiply \sqrt{5} and \sqrt{7}, multiply the numbers under the square root.

$$14+6\sqrt{2}-\left(\sqrt{5}-\sqrt{35}+1-\sqrt{7}\right)$$

To find the opposite of \sqrt{5}-\sqrt{35}+1-\sqrt{7}, find the opposite of each term.

$$14+6\sqrt{2}-\sqrt{5}+\sqrt{35}-1+\sqrt{7}$$

Subtract 1 from 14 to get 13.

$$13+6\sqrt{2}-\sqrt{5}+\sqrt{35}+\sqrt{7}$$