Solve =(2+sqrt{2}+sqrt{3})(2+sqrt{2}-sqrt{3})

$$= ( 2 + \sqrt { 2 } + \sqrt { 3 } ) ( 2 + \sqrt { 2 } - \sqrt { 3 } )$$

Evaluate

$$4\sqrt{2}+3\approx 8.656854249$$

Solution Steps

Apply the distributive property by multiplying each term of 2+\sqrt{2}+\sqrt{3} by each term of 2+\sqrt{2}-\sqrt{3}.

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

Combine 2\sqrt{2} and 2\sqrt{2} to get 4\sqrt{2}.

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

The square of \sqrt{2} is 2.

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

Add 4 and 2 to get 6.

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

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

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

Combine -2\sqrt{3} and 2\sqrt{3} to get 0.

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

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

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

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

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

The square of \sqrt{3} is 3.

$$6+4\sqrt{2}-3$$

Subtract 3 from 6 to get 3.

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