Solve sqrt{5}(sqrt{3}-2sqrt{2})+(sqrt{2}+sqrt{5})^2-sqrt{60}(-1-sqrt{15})

$$\sqrt { 5 } ( \sqrt { 3 } - 2 \sqrt { 2 } ) + ( \sqrt { 2 } + \sqrt { 5 } ) ^ { 2 } - \sqrt { 60 } ( - 1 - \sqrt { 15 } )$$

Evaluate

$$3\sqrt{15}+37\approx 48.618950039$$

Solution Steps

Use the distributive property to multiply \sqrt{5} by \sqrt{3}-2\sqrt{2}.

$$\sqrt{5}\sqrt{3}-2\sqrt{5}\sqrt{2}+\left(\sqrt{2}+\sqrt{5}\right)^{2}-\sqrt{60}\left(-1-\sqrt{15}\right)$$

To multiply \sqrt{5} and \sqrt{3}, multiply the numbers under the square root.

$$\sqrt{15}-2\sqrt{5}\sqrt{2}+\left(\sqrt{2}+\sqrt{5}\right)^{2}-\sqrt{60}\left(-1-\sqrt{15}\right)$$

To multiply \sqrt{5} and \sqrt{2}, multiply the numbers under the square root.

$$\sqrt{15}-2\sqrt{10}+\left(\sqrt{2}+\sqrt{5}\right)^{2}-\sqrt{60}\left(-1-\sqrt{15}\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{2}+\sqrt{5}\right)^{2}.

$$\sqrt{15}-2\sqrt{10}+\left(\sqrt{2}\right)^{2}+2\sqrt{2}\sqrt{5}+\left(\sqrt{5}\right)^{2}-\sqrt{60}\left(-1-\sqrt{15}\right)$$

The square of \sqrt{2} is 2.

$$\sqrt{15}-2\sqrt{10}+2+2\sqrt{2}\sqrt{5}+\left(\sqrt{5}\right)^{2}-\sqrt{60}\left(-1-\sqrt{15}\right)$$

To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.

$$\sqrt{15}-2\sqrt{10}+2+2\sqrt{10}+\left(\sqrt{5}\right)^{2}-\sqrt{60}\left(-1-\sqrt{15}\right)$$

The square of \sqrt{5} is 5.

$$\sqrt{15}-2\sqrt{10}+2+2\sqrt{10}+5-\sqrt{60}\left(-1-\sqrt{15}\right)$$

Add 2 and 5 to get 7.

$$\sqrt{15}-2\sqrt{10}+7+2\sqrt{10}-\sqrt{60}\left(-1-\sqrt{15}\right)$$

Combine -2\sqrt{10} and 2\sqrt{10} to get 0.

$$\sqrt{15}+7-\sqrt{60}\left(-1-\sqrt{15}\right)$$

Factor 60=2^{2}\times 15. Rewrite the square root of the product \sqrt{2^{2}\times 15} as the product of square roots \sqrt{2^{2}}\sqrt{15}. Take the square root of 2^{2}.

$$\sqrt{15}+7-2\sqrt{15}\left(-1-\sqrt{15}\right)$$

Use the distributive property to multiply -2\sqrt{15} by -1-\sqrt{15}.

$$\sqrt{15}+7+2\sqrt{15}+2\left(\sqrt{15}\right)^{2}$$

The square of \sqrt{15} is 15.

$$\sqrt{15}+7+2\sqrt{15}+2\times 15$$

Multiply 2 and 15 to get 30.

$$\sqrt{15}+7+2\sqrt{15}+30$$

Combine \sqrt{15} and 2\sqrt{15} to get 3\sqrt{15}.

$$3\sqrt{15}+7+30$$

Add 7 and 30 to get 37.

$$3\sqrt{15}+37$$