Solve (sqrt{2}+sqrt{3}-1)(sqrt{2}-sqrt{3}+1)

$$( \sqrt{ 2 } + \sqrt{ 3 } -1)( \sqrt{ 2 } - \sqrt{ 3 } +1)$$

Evaluate

$$2\left(\sqrt{3}-1\right)\approx 1.464101615$$

Solution Steps

Apply the distributive property by multiplying each term of \sqrt{2}+\sqrt{3}-1 by each term of \sqrt{2}-\sqrt{3}+1.

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

The square of \sqrt{2} is 2.

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

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

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

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

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

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

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

The square of \sqrt{3} is 3.

$$2+\sqrt{2}-3+\sqrt{3}-\sqrt{2}+\sqrt{3}-1$$

Subtract 3 from 2 to get -1.

$$-1+\sqrt{2}+\sqrt{3}-\sqrt{2}+\sqrt{3}-1$$

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

$$-1+\sqrt{3}+\sqrt{3}-1$$

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

$$-1+2\sqrt{3}-1$$

Subtract 1 from -1 to get -2.

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