Solve (2+2sqrt{7})^2+(4sqrt{7}-2)^2

$$( 2 + 2 \sqrt { 7 } ) ^ { 2 } + ( 4 \sqrt { 7 } - 2 ) ^ { 2 }$$

Evaluate

$$148-8\sqrt{7}\approx 126.833989511$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2+2\sqrt{7}\right)^{2}.

$$4+8\sqrt{7}+4\left(\sqrt{7}\right)^{2}+\left(4\sqrt{7}-2\right)^{2}$$

The square of \sqrt{7} is 7.

$$4+8\sqrt{7}+4\times 7+\left(4\sqrt{7}-2\right)^{2}$$

Multiply 4 and 7 to get 28.

$$4+8\sqrt{7}+28+\left(4\sqrt{7}-2\right)^{2}$$

Add 4 and 28 to get 32.

$$32+8\sqrt{7}+\left(4\sqrt{7}-2\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4\sqrt{7}-2\right)^{2}.

$$32+8\sqrt{7}+16\left(\sqrt{7}\right)^{2}-16\sqrt{7}+4$$

The square of \sqrt{7} is 7.

$$32+8\sqrt{7}+16\times 7-16\sqrt{7}+4$$

Multiply 16 and 7 to get 112.

$$32+8\sqrt{7}+112-16\sqrt{7}+4$$

Add 112 and 4 to get 116.

$$32+8\sqrt{7}+116-16\sqrt{7}$$

Add 32 and 116 to get 148.

$$148+8\sqrt{7}-16\sqrt{7}$$

Combine 8\sqrt{7} and -16\sqrt{7} to get -8\sqrt{7}.

$$148-8\sqrt{7}$$

Expand

$$148-8\sqrt{7}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2+2\sqrt{7}\right)^{2}.

$$4+8\sqrt{7}+4\left(\sqrt{7}\right)^{2}+\left(4\sqrt{7}-2\right)^{2}$$

The square of \sqrt{7} is 7.

$$4+8\sqrt{7}+4\times 7+\left(4\sqrt{7}-2\right)^{2}$$

Multiply 4 and 7 to get 28.

$$4+8\sqrt{7}+28+\left(4\sqrt{7}-2\right)^{2}$$

Add 4 and 28 to get 32.

$$32+8\sqrt{7}+\left(4\sqrt{7}-2\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4\sqrt{7}-2\right)^{2}.

$$32+8\sqrt{7}+16\left(\sqrt{7}\right)^{2}-16\sqrt{7}+4$$

The square of \sqrt{7} is 7.

$$32+8\sqrt{7}+16\times 7-16\sqrt{7}+4$$

Multiply 16 and 7 to get 112.

$$32+8\sqrt{7}+112-16\sqrt{7}+4$$

Add 112 and 4 to get 116.

$$32+8\sqrt{7}+116-16\sqrt{7}$$

Add 32 and 116 to get 148.

$$148+8\sqrt{7}-16\sqrt{7}$$

Combine 8\sqrt{7} and -16\sqrt{7} to get -8\sqrt{7}.

$$148-8\sqrt{7}$$