Solve (2+2sqrt{3})^2+(2+2sqrt{3})^2=?

$$( 2 + 2 \sqrt { 3 } ) ^ { 2 } + ( 2 + 2 \sqrt { 3 } ) ^ { 2 } = ?$$

Evaluate

$$16\left(\sqrt{3}+2\right)\approx 59.712812921$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2+2\sqrt{3}\right)^{2}.

$$4+8\sqrt{3}+4\left(\sqrt{3}\right)^{2}+\left(2+2\sqrt{3}\right)^{2}$$

The square of \sqrt{3} is 3.

$$4+8\sqrt{3}+4\times 3+\left(2+2\sqrt{3}\right)^{2}$$

Multiply 4 and 3 to get 12.

$$4+8\sqrt{3}+12+\left(2+2\sqrt{3}\right)^{2}$$

Add 4 and 12 to get 16.

$$16+8\sqrt{3}+\left(2+2\sqrt{3}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2+2\sqrt{3}\right)^{2}.

$$16+8\sqrt{3}+4+8\sqrt{3}+4\left(\sqrt{3}\right)^{2}$$

The square of \sqrt{3} is 3.

$$16+8\sqrt{3}+4+8\sqrt{3}+4\times 3$$

Multiply 4 and 3 to get 12.

$$16+8\sqrt{3}+4+8\sqrt{3}+12$$

Add 4 and 12 to get 16.

$$16+8\sqrt{3}+16+8\sqrt{3}$$

Add 16 and 16 to get 32.

$$32+8\sqrt{3}+8\sqrt{3}$$

Combine 8\sqrt{3} and 8\sqrt{3} to get 16\sqrt{3}.

$$32+16\sqrt{3}$$

Expand

$$16 \sqrt{3} + 32 = 59.712812921$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2+2\sqrt{3}\right)^{2}.

$$4+8\sqrt{3}+4\left(\sqrt{3}\right)^{2}+\left(2+2\sqrt{3}\right)^{2}$$

The square of \sqrt{3} is 3.

$$4+8\sqrt{3}+4\times 3+\left(2+2\sqrt{3}\right)^{2}$$

Multiply 4 and 3 to get 12.

$$4+8\sqrt{3}+12+\left(2+2\sqrt{3}\right)^{2}$$

Add 4 and 12 to get 16.

$$16+8\sqrt{3}+\left(2+2\sqrt{3}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2+2\sqrt{3}\right)^{2}.

$$16+8\sqrt{3}+4+8\sqrt{3}+4\left(\sqrt{3}\right)^{2}$$

The square of \sqrt{3} is 3.

$$16+8\sqrt{3}+4+8\sqrt{3}+4\times 3$$

Multiply 4 and 3 to get 12.

$$16+8\sqrt{3}+4+8\sqrt{3}+12$$

Add 4 and 12 to get 16.

$$16+8\sqrt{3}+16+8\sqrt{3}$$

Add 16 and 16 to get 32.

$$32+8\sqrt{3}+8\sqrt{3}$$

Combine 8\sqrt{3} and 8\sqrt{3} to get 16\sqrt{3}.

$$32+16\sqrt{3}$$