Solve (8sqrt{2}-4sqrt{6})div2sqrt{2}-(sqrt{3}-1)^2

$$( 8 \sqrt { 2 } - 4 \sqrt { 6 } ) \div 2 \sqrt { 2 } - ( \sqrt { 3 } - 1 ) ^ { 2 }$$

Evaluate

$$4-2\sqrt{3}\approx 0.535898385$$

Short Solution Steps

Express \frac{8\sqrt{2}-4\sqrt{6}}{2}\sqrt{2} as a single fraction.

$$\frac{\left(8\sqrt{2}-4\sqrt{6}\right)\sqrt{2}}{2}-\left(\sqrt{3}-1\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{3}-1\right)^{2}.

$$\frac{\left(8\sqrt{2}-4\sqrt{6}\right)\sqrt{2}}{2}-\left(\left(\sqrt{3}\right)^{2}-2\sqrt{3}+1\right)$$

The square of \sqrt{3} is 3.

$$\frac{\left(8\sqrt{2}-4\sqrt{6}\right)\sqrt{2}}{2}-\left(3-2\sqrt{3}+1\right)$$

Add 3 and 1 to get 4.

$$\frac{\left(8\sqrt{2}-4\sqrt{6}\right)\sqrt{2}}{2}-\left(4-2\sqrt{3}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 4-2\sqrt{3} times \frac{2}{2}.

$$\frac{\left(8\sqrt{2}-4\sqrt{6}\right)\sqrt{2}}{2}-\frac{2\left(4-2\sqrt{3}\right)}{2}$$

Since \frac{\left(8\sqrt{2}-4\sqrt{6}\right)\sqrt{2}}{2} and \frac{2\left(4-2\sqrt{3}\right)}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(8\sqrt{2}-4\sqrt{6}\right)\sqrt{2}-2\left(4-2\sqrt{3}\right)}{2}$$

Do the multiplications in \left(8\sqrt{2}-4\sqrt{6}\right)\sqrt{2}-2\left(4-2\sqrt{3}\right).

$$\frac{16-8\sqrt{3}-8+4\sqrt{3}}{2}$$

Do the calculations in 16-8\sqrt{3}-8+4\sqrt{3}.

$$\frac{8-4\sqrt{3}}{2}$$