Solve (10sqrt{6}+10sqrt{2})^2

$$( 10 \sqrt { 6 } + 10 \sqrt { 2 } ) ^ { 2 }$$

Evaluate

$$400\left(\sqrt{3}+2\right)\approx 1492.820323028$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(10\sqrt{6}+10\sqrt{2}\right)^{2}.

$$100\left(\sqrt{6}\right)^{2}+200\sqrt{6}\sqrt{2}+100\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{6} is 6.

$$100\times 6+200\sqrt{6}\sqrt{2}+100\left(\sqrt{2}\right)^{2}$$

Multiply 100 and 6 to get 600.

$$600+200\sqrt{6}\sqrt{2}+100\left(\sqrt{2}\right)^{2}$$

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}.

$$600+200\sqrt{2}\sqrt{3}\sqrt{2}+100\left(\sqrt{2}\right)^{2}$$

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

$$600+200\times 2\sqrt{3}+100\left(\sqrt{2}\right)^{2}$$

Multiply 200 and 2 to get 400.

$$600+400\sqrt{3}+100\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$600+400\sqrt{3}+100\times 2$$

Multiply 100 and 2 to get 200.

$$600+400\sqrt{3}+200$$

Add 600 and 200 to get 800.

$$800+400\sqrt{3}$$

Expand

$$400 \sqrt{3} + 800 = 1492.820323028$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(10\sqrt{6}+10\sqrt{2}\right)^{2}.

$$100\left(\sqrt{6}\right)^{2}+200\sqrt{6}\sqrt{2}+100\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{6} is 6.

$$100\times 6+200\sqrt{6}\sqrt{2}+100\left(\sqrt{2}\right)^{2}$$

Multiply 100 and 6 to get 600.

$$600+200\sqrt{6}\sqrt{2}+100\left(\sqrt{2}\right)^{2}$$

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}.

$$600+200\sqrt{2}\sqrt{3}\sqrt{2}+100\left(\sqrt{2}\right)^{2}$$

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

$$600+200\times 2\sqrt{3}+100\left(\sqrt{2}\right)^{2}$$

Multiply 200 and 2 to get 400.

$$600+400\sqrt{3}+100\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$600+400\sqrt{3}+100\times 2$$

Multiply 100 and 2 to get 200.

$$600+400\sqrt{3}+200$$

Add 600 and 200 to get 800.

$$800+400\sqrt{3}$$