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

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

Evaluate

$$4\left(\sqrt{6}-2\right)\approx 1.797958971$$

Solution Steps

Consider \left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{5}+\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\left(\sqrt{5}\right)^{2}-\left(\sqrt{3}\right)^{2}-\left(2-\sqrt{6}\right)^{2}$$

The square of \sqrt{5} is 5.

$$5-\left(\sqrt{3}\right)^{2}-\left(2-\sqrt{6}\right)^{2}$$

The square of \sqrt{3} is 3.

$$5-3-\left(2-\sqrt{6}\right)^{2}$$

Subtract 3 from 5 to get 2.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2-\sqrt{6}\right)^{2}.

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

The square of \sqrt{6} is 6.

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

Add 4 and 6 to get 10.

$$2-\left(10-4\sqrt{6}\right)$$

To find the opposite of 10-4\sqrt{6}, find the opposite of each term.

$$2-10+4\sqrt{6}$$

Subtract 10 from 2 to get -8.

$$-8+4\sqrt{6}$$