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

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

Evaluate

$$1-2\sqrt{2}-2\sqrt{3}\approx -5.29252874$$

Solution Steps

Apply the distributive property by multiplying each term of 1+\sqrt{3} by each term of \sqrt{2}-\sqrt{6}.

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

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

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

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

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

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}.

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

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

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

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

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

To find the opposite of 2\sqrt{3}-1, find the opposite of each term.

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

The opposite of -1 is 1.

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