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

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

Evaluate

$$5\left(\sqrt{3}+4\right)\approx 28.660254038$$

Solution Steps

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

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

The square of \sqrt{5} is 5.

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

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}.

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

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

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

Use the distributive property to multiply \sqrt{15}+2\sqrt{5} by \sqrt{15}.

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

The square of \sqrt{15} is 15.

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

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}.

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

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

$$5-5\sqrt{3}+15+2\times 5\sqrt{3}$$

Multiply 2 and 5 to get 10.

$$5-5\sqrt{3}+15+10\sqrt{3}$$

Add 5 and 15 to get 20.

$$20-5\sqrt{3}+10\sqrt{3}$$

Combine -5\sqrt{3} and 10\sqrt{3} to get 5\sqrt{3}.

$$20+5\sqrt{3}$$