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

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

Evaluate

$$2\left(\sqrt{10}+5-\sqrt{6}-\sqrt{15}\right)\approx 3.679609142$$

Solution Steps

Square \sqrt{5}-\sqrt{3}+\sqrt{2}.

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

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

$$2\sqrt{10}-2\sqrt{2}\sqrt{3}-2\sqrt{3}\sqrt{5}+\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}\right)^{2}+\left(\sqrt{5}\right)^{2}$$

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

$$2\sqrt{10}-2\sqrt{6}-2\sqrt{3}\sqrt{5}+\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}\right)^{2}+\left(\sqrt{5}\right)^{2}$$

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

$$2\sqrt{10}-2\sqrt{6}-2\sqrt{15}+\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}\right)^{2}+\left(\sqrt{5}\right)^{2}$$

The square of \sqrt{2} is 2.

$$2\sqrt{10}-2\sqrt{6}-2\sqrt{15}+2+\left(\sqrt{3}\right)^{2}+\left(\sqrt{5}\right)^{2}$$

The square of \sqrt{3} is 3.

$$2\sqrt{10}-2\sqrt{6}-2\sqrt{15}+2+3+\left(\sqrt{5}\right)^{2}$$

Add 2 and 3 to get 5.

$$2\sqrt{10}-2\sqrt{6}-2\sqrt{15}+5+\left(\sqrt{5}\right)^{2}$$

The square of \sqrt{5} is 5.

$$2\sqrt{10}-2\sqrt{6}-2\sqrt{15}+5+5$$

Add 5 and 5 to get 10.

$$2\sqrt{10}-2\sqrt{6}-2\sqrt{15}+10$$

Expand

$$2 \sqrt{10} + 10 - 2 \sqrt{6} - 2 \sqrt{15} = 3.679609142$$

Solution Steps

Square \sqrt{5}-\sqrt{3}+\sqrt{2}.

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

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

$$2\sqrt{10}-2\sqrt{2}\sqrt{3}-2\sqrt{3}\sqrt{5}+\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}\right)^{2}+\left(\sqrt{5}\right)^{2}$$

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

$$2\sqrt{10}-2\sqrt{6}-2\sqrt{3}\sqrt{5}+\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}\right)^{2}+\left(\sqrt{5}\right)^{2}$$

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

$$2\sqrt{10}-2\sqrt{6}-2\sqrt{15}+\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}\right)^{2}+\left(\sqrt{5}\right)^{2}$$

The square of \sqrt{2} is 2.

$$2\sqrt{10}-2\sqrt{6}-2\sqrt{15}+2+\left(\sqrt{3}\right)^{2}+\left(\sqrt{5}\right)^{2}$$

The square of \sqrt{3} is 3.

$$2\sqrt{10}-2\sqrt{6}-2\sqrt{15}+2+3+\left(\sqrt{5}\right)^{2}$$

Add 2 and 3 to get 5.

$$2\sqrt{10}-2\sqrt{6}-2\sqrt{15}+5+\left(\sqrt{5}\right)^{2}$$

The square of \sqrt{5} is 5.

$$2\sqrt{10}-2\sqrt{6}-2\sqrt{15}+5+5$$

Add 5 and 5 to get 10.

$$2\sqrt{10}-2\sqrt{6}-2\sqrt{15}+10$$