Solve {left(sqrt{3}+sqrt{2}+sqrt{6}right)}^2

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

Evaluate

$$2\sqrt{6}+4\sqrt{3}+6\sqrt{2}+11\approx 31.31246409$$

Solution Steps

Square \sqrt{3}+\sqrt{2}+\sqrt{6}.

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

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

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

Factor 6=2\times 3. Rewrite the square root of the product \sqrt{2\times 3} as the product of square roots \sqrt{2}\sqrt{3}.

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

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$2\sqrt{6}+2\times 2\sqrt{3}+2\sqrt{3}\sqrt{6}+\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}$$

Multiply 2 and 2 to get 4.

$$2\sqrt{6}+4\sqrt{3}+2\sqrt{3}\sqrt{6}+\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}$$

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

$$2\sqrt{6}+4\sqrt{3}+2\sqrt{3}\sqrt{3}\sqrt{2}+\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$2\sqrt{6}+4\sqrt{3}+2\times 3\sqrt{2}+\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}$$

Multiply 2 and 3 to get 6.

$$2\sqrt{6}+4\sqrt{3}+6\sqrt{2}+\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}$$

The square of \sqrt{2} is 2.

$$2\sqrt{6}+4\sqrt{3}+6\sqrt{2}+2+\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}$$

The square of \sqrt{3} is 3.

$$2\sqrt{6}+4\sqrt{3}+6\sqrt{2}+2+3+\left(\sqrt{6}\right)^{2}$$

Add 2 and 3 to get 5.

$$2\sqrt{6}+4\sqrt{3}+6\sqrt{2}+5+\left(\sqrt{6}\right)^{2}$$

The square of \sqrt{6} is 6.

$$2\sqrt{6}+4\sqrt{3}+6\sqrt{2}+5+6$$

Add 5 and 6 to get 11.

$$2\sqrt{6}+4\sqrt{3}+6\sqrt{2}+11$$

Expand

$$2 \sqrt{6} + 4 \sqrt{3} + 6 \sqrt{2} + 11 = 31.31246409$$

Solution Steps

Square \sqrt{3}+\sqrt{2}+\sqrt{6}.

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

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

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

Factor 6=2\times 3. Rewrite the square root of the product \sqrt{2\times 3} as the product of square roots \sqrt{2}\sqrt{3}.

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

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$2\sqrt{6}+2\times 2\sqrt{3}+2\sqrt{3}\sqrt{6}+\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}$$

Multiply 2 and 2 to get 4.

$$2\sqrt{6}+4\sqrt{3}+2\sqrt{3}\sqrt{6}+\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}$$

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

$$2\sqrt{6}+4\sqrt{3}+2\sqrt{3}\sqrt{3}\sqrt{2}+\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$2\sqrt{6}+4\sqrt{3}+2\times 3\sqrt{2}+\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}$$

Multiply 2 and 3 to get 6.

$$2\sqrt{6}+4\sqrt{3}+6\sqrt{2}+\left(\sqrt{2}\right)^{2}+\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}$$

The square of \sqrt{2} is 2.

$$2\sqrt{6}+4\sqrt{3}+6\sqrt{2}+2+\left(\sqrt{3}\right)^{2}+\left(\sqrt{6}\right)^{2}$$

The square of \sqrt{3} is 3.

$$2\sqrt{6}+4\sqrt{3}+6\sqrt{2}+2+3+\left(\sqrt{6}\right)^{2}$$

Add 2 and 3 to get 5.

$$2\sqrt{6}+4\sqrt{3}+6\sqrt{2}+5+\left(\sqrt{6}\right)^{2}$$

The square of \sqrt{6} is 6.

$$2\sqrt{6}+4\sqrt{3}+6\sqrt{2}+5+6$$

Add 5 and 6 to get 11.

$$2\sqrt{6}+4\sqrt{3}+6\sqrt{2}+11$$