Solve sqrt{5}-sqrt{15}+left(sqrt{15}+2sqrt{3}right)left(sqrt{15}-2sqrt{3}right)

$$\sqrt{ 5 } - \sqrt{ 15 } + \left( \sqrt{ 15 } +2 \sqrt{ 3 } \right) \left( \sqrt{ 15 } -2 \sqrt{ 3 } \right)$$

Evaluate

$$\sqrt{5}+3-\sqrt{15}\approx 1.363084631$$

Solution Steps

Consider \left(\sqrt{15}+2\sqrt{3}\right)\left(\sqrt{15}-2\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}.

$$\sqrt{5}-\sqrt{15}+\left(\sqrt{15}\right)^{2}-\left(2\sqrt{3}\right)^{2}$$

The square of \sqrt{15} is 15.

$$\sqrt{5}-\sqrt{15}+15-\left(2\sqrt{3}\right)^{2}$$

Expand \left(2\sqrt{3}\right)^{2}.

$$\sqrt{5}-\sqrt{15}+15-2^{2}\left(\sqrt{3}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$\sqrt{5}-\sqrt{15}+15-4\left(\sqrt{3}\right)^{2}$$

The square of \sqrt{3} is 3.

$$\sqrt{5}-\sqrt{15}+15-4\times 3$$

Multiply 4 and 3 to get 12.

$$\sqrt{5}-\sqrt{15}+15-12$$

Subtract 12 from 15 to get 3.

$$\sqrt{5}-\sqrt{15}+3$$