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

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

Evaluate

$$8-5\sqrt{3}\approx -0.660254038$$

Solution Steps

Use the distributive property to multiply \sqrt{5} by \sqrt{5}-\sqrt{15}.

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

The square of \sqrt{5} is 5.

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

Factor 15=5\times 3. Rewrite the square root of the product \sqrt{5\times 3} as the product of square roots \sqrt{5}\sqrt{3}.

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

Multiply \sqrt{5} and \sqrt{5} to get 5.

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

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}.

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

The square of \sqrt{15} is 15.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{3} is 3.

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

Multiply 4 and 3 to get 12.

$$5-5\sqrt{3}+15-12$$

Subtract 12 from 15 to get 3.

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

Add 5 and 3 to get 8.

$$8-5\sqrt{3}$$