Solve (2+2sqrt{5}+2sqrt{2})*(2+2sqrt{5}-2sqrt{2})*(2+2sqrt{2}-2sqrt{5})*(2sqrt{5}+2sqrt{2}-2)

$$(2+2 \sqrt{ 5 } +2 \sqrt{ 2 } ) \times (2+2 \sqrt{ 5 } -2 \sqrt{ 2 } ) \times (2+2 \sqrt{ 2 } -2 \sqrt{ 5 } ) \times (2 \sqrt{ 5 } +2 \sqrt{ 2 } -2)$$

Evaluate

$$64$$

Solution Steps

Apply the distributive property by multiplying each term of 2+2\sqrt{5}+2\sqrt{2} by each term of 2+2\sqrt{5}-2\sqrt{2}.

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

Combine 4\sqrt{5} and 4\sqrt{5} to get 8\sqrt{5}.

$$\left(4+8\sqrt{5}-4\sqrt{2}+4\left(\sqrt{5}\right)^{2}-4\sqrt{5}\sqrt{2}+4\sqrt{2}+4\sqrt{2}\sqrt{5}-4\left(\sqrt{2}\right)^{2}\right)\left(2+2\sqrt{2}-2\sqrt{5}\right)\left(2\sqrt{5}+2\sqrt{2}-2\right)$$

The square of \sqrt{5} is 5.

$$\left(4+8\sqrt{5}-4\sqrt{2}+4\times 5-4\sqrt{5}\sqrt{2}+4\sqrt{2}+4\sqrt{2}\sqrt{5}-4\left(\sqrt{2}\right)^{2}\right)\left(2+2\sqrt{2}-2\sqrt{5}\right)\left(2\sqrt{5}+2\sqrt{2}-2\right)$$

Multiply 4 and 5 to get 20.

$$\left(4+8\sqrt{5}-4\sqrt{2}+20-4\sqrt{5}\sqrt{2}+4\sqrt{2}+4\sqrt{2}\sqrt{5}-4\left(\sqrt{2}\right)^{2}\right)\left(2+2\sqrt{2}-2\sqrt{5}\right)\left(2\sqrt{5}+2\sqrt{2}-2\right)$$

Add 4 and 20 to get 24.

$$\left(24+8\sqrt{5}-4\sqrt{2}-4\sqrt{5}\sqrt{2}+4\sqrt{2}+4\sqrt{2}\sqrt{5}-4\left(\sqrt{2}\right)^{2}\right)\left(2+2\sqrt{2}-2\sqrt{5}\right)\left(2\sqrt{5}+2\sqrt{2}-2\right)$$

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

$$\left(24+8\sqrt{5}-4\sqrt{2}-4\sqrt{10}+4\sqrt{2}+4\sqrt{2}\sqrt{5}-4\left(\sqrt{2}\right)^{2}\right)\left(2+2\sqrt{2}-2\sqrt{5}\right)\left(2\sqrt{5}+2\sqrt{2}-2\right)$$

Combine -4\sqrt{2} and 4\sqrt{2} to get 0.

$$\left(24+8\sqrt{5}-4\sqrt{10}+4\sqrt{2}\sqrt{5}-4\left(\sqrt{2}\right)^{2}\right)\left(2+2\sqrt{2}-2\sqrt{5}\right)\left(2\sqrt{5}+2\sqrt{2}-2\right)$$

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

$$\left(24+8\sqrt{5}-4\sqrt{10}+4\sqrt{10}-4\left(\sqrt{2}\right)^{2}\right)\left(2+2\sqrt{2}-2\sqrt{5}\right)\left(2\sqrt{5}+2\sqrt{2}-2\right)$$

Combine -4\sqrt{10} and 4\sqrt{10} to get 0.

$$\left(24+8\sqrt{5}-4\left(\sqrt{2}\right)^{2}\right)\left(2+2\sqrt{2}-2\sqrt{5}\right)\left(2\sqrt{5}+2\sqrt{2}-2\right)$$

The square of \sqrt{2} is 2.

$$\left(24+8\sqrt{5}-4\times 2\right)\left(2+2\sqrt{2}-2\sqrt{5}\right)\left(2\sqrt{5}+2\sqrt{2}-2\right)$$

Multiply -4 and 2 to get -8.

$$\left(24+8\sqrt{5}-8\right)\left(2+2\sqrt{2}-2\sqrt{5}\right)\left(2\sqrt{5}+2\sqrt{2}-2\right)$$

Subtract 8 from 24 to get 16.

$$\left(16+8\sqrt{5}\right)\left(2+2\sqrt{2}-2\sqrt{5}\right)\left(2\sqrt{5}+2\sqrt{2}-2\right)$$

Apply the distributive property by multiplying each term of 16+8\sqrt{5} by each term of 2+2\sqrt{2}-2\sqrt{5}.

$$\left(32+32\sqrt{2}-32\sqrt{5}+16\sqrt{5}+16\sqrt{5}\sqrt{2}-16\left(\sqrt{5}\right)^{2}\right)\left(2\sqrt{5}+2\sqrt{2}-2\right)$$

Combine -32\sqrt{5} and 16\sqrt{5} to get -16\sqrt{5}.

$$\left(32+32\sqrt{2}-16\sqrt{5}+16\sqrt{5}\sqrt{2}-16\left(\sqrt{5}\right)^{2}\right)\left(2\sqrt{5}+2\sqrt{2}-2\right)$$

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

$$\left(32+32\sqrt{2}-16\sqrt{5}+16\sqrt{10}-16\left(\sqrt{5}\right)^{2}\right)\left(2\sqrt{5}+2\sqrt{2}-2\right)$$

The square of \sqrt{5} is 5.

$$\left(32+32\sqrt{2}-16\sqrt{5}+16\sqrt{10}-16\times 5\right)\left(2\sqrt{5}+2\sqrt{2}-2\right)$$

Multiply -16 and 5 to get -80.

$$\left(32+32\sqrt{2}-16\sqrt{5}+16\sqrt{10}-80\right)\left(2\sqrt{5}+2\sqrt{2}-2\right)$$

Subtract 80 from 32 to get -48.

$$\left(-48+32\sqrt{2}-16\sqrt{5}+16\sqrt{10}\right)\left(2\sqrt{5}+2\sqrt{2}-2\right)$$

Apply the distributive property by multiplying each term of -48+32\sqrt{2}-16\sqrt{5}+16\sqrt{10} by each term of 2\sqrt{5}+2\sqrt{2}-2.

$$-96\sqrt{5}-96\sqrt{2}+96+64\sqrt{2}\sqrt{5}+64\left(\sqrt{2}\right)^{2}-64\sqrt{2}-32\left(\sqrt{5}\right)^{2}-32\sqrt{5}\sqrt{2}+32\sqrt{5}+32\sqrt{10}\sqrt{5}+32\sqrt{2}\sqrt{10}-32\sqrt{10}$$

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

$$-96\sqrt{5}-96\sqrt{2}+96+64\sqrt{10}+64\left(\sqrt{2}\right)^{2}-64\sqrt{2}-32\left(\sqrt{5}\right)^{2}-32\sqrt{5}\sqrt{2}+32\sqrt{5}+32\sqrt{10}\sqrt{5}+32\sqrt{2}\sqrt{10}-32\sqrt{10}$$

The square of \sqrt{2} is 2.

$$-96\sqrt{5}-96\sqrt{2}+96+64\sqrt{10}+64\times 2-64\sqrt{2}-32\left(\sqrt{5}\right)^{2}-32\sqrt{5}\sqrt{2}+32\sqrt{5}+32\sqrt{10}\sqrt{5}+32\sqrt{2}\sqrt{10}-32\sqrt{10}$$

Multiply 64 and 2 to get 128.

$$-96\sqrt{5}-96\sqrt{2}+96+64\sqrt{10}+128-64\sqrt{2}-32\left(\sqrt{5}\right)^{2}-32\sqrt{5}\sqrt{2}+32\sqrt{5}+32\sqrt{10}\sqrt{5}+32\sqrt{2}\sqrt{10}-32\sqrt{10}$$

Add 96 and 128 to get 224.

$$-96\sqrt{5}-96\sqrt{2}+224+64\sqrt{10}-64\sqrt{2}-32\left(\sqrt{5}\right)^{2}-32\sqrt{5}\sqrt{2}+32\sqrt{5}+32\sqrt{10}\sqrt{5}+32\sqrt{2}\sqrt{10}-32\sqrt{10}$$

Combine -96\sqrt{2} and -64\sqrt{2} to get -160\sqrt{2}.

$$-96\sqrt{5}-160\sqrt{2}+224+64\sqrt{10}-32\left(\sqrt{5}\right)^{2}-32\sqrt{5}\sqrt{2}+32\sqrt{5}+32\sqrt{10}\sqrt{5}+32\sqrt{2}\sqrt{10}-32\sqrt{10}$$

The square of \sqrt{5} is 5.

$$-96\sqrt{5}-160\sqrt{2}+224+64\sqrt{10}-32\times 5-32\sqrt{5}\sqrt{2}+32\sqrt{5}+32\sqrt{10}\sqrt{5}+32\sqrt{2}\sqrt{10}-32\sqrt{10}$$

Multiply -32 and 5 to get -160.

$$-96\sqrt{5}-160\sqrt{2}+224+64\sqrt{10}-160-32\sqrt{5}\sqrt{2}+32\sqrt{5}+32\sqrt{10}\sqrt{5}+32\sqrt{2}\sqrt{10}-32\sqrt{10}$$

Subtract 160 from 224 to get 64.

$$-96\sqrt{5}-160\sqrt{2}+64+64\sqrt{10}-32\sqrt{5}\sqrt{2}+32\sqrt{5}+32\sqrt{10}\sqrt{5}+32\sqrt{2}\sqrt{10}-32\sqrt{10}$$

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

$$-96\sqrt{5}-160\sqrt{2}+64+64\sqrt{10}-32\sqrt{10}+32\sqrt{5}+32\sqrt{10}\sqrt{5}+32\sqrt{2}\sqrt{10}-32\sqrt{10}$$

Combine 64\sqrt{10} and -32\sqrt{10} to get 32\sqrt{10}.

$$-96\sqrt{5}-160\sqrt{2}+64+32\sqrt{10}+32\sqrt{5}+32\sqrt{10}\sqrt{5}+32\sqrt{2}\sqrt{10}-32\sqrt{10}$$

Combine -96\sqrt{5} and 32\sqrt{5} to get -64\sqrt{5}.

$$-64\sqrt{5}-160\sqrt{2}+64+32\sqrt{10}+32\sqrt{10}\sqrt{5}+32\sqrt{2}\sqrt{10}-32\sqrt{10}$$

Factor 10=5\times 2. Rewrite the square root of the product \sqrt{5\times 2} as the product of square roots \sqrt{5}\sqrt{2}.

$$-64\sqrt{5}-160\sqrt{2}+64+32\sqrt{10}+32\sqrt{5}\sqrt{2}\sqrt{5}+32\sqrt{2}\sqrt{10}-32\sqrt{10}$$

Multiply \sqrt{5} and \sqrt{5} to get 5.

$$-64\sqrt{5}-160\sqrt{2}+64+32\sqrt{10}+32\times 5\sqrt{2}+32\sqrt{2}\sqrt{10}-32\sqrt{10}$$

Multiply 32 and 5 to get 160.

$$-64\sqrt{5}-160\sqrt{2}+64+32\sqrt{10}+160\sqrt{2}+32\sqrt{2}\sqrt{10}-32\sqrt{10}$$

Combine -160\sqrt{2} and 160\sqrt{2} to get 0.

$$-64\sqrt{5}+64+32\sqrt{10}+32\sqrt{2}\sqrt{10}-32\sqrt{10}$$

Factor 10=2\times 5. Rewrite the square root of the product \sqrt{2\times 5} as the product of square roots \sqrt{2}\sqrt{5}.

$$-64\sqrt{5}+64+32\sqrt{10}+32\sqrt{2}\sqrt{2}\sqrt{5}-32\sqrt{10}$$

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

$$-64\sqrt{5}+64+32\sqrt{10}+32\times 2\sqrt{5}-32\sqrt{10}$$

Multiply 32 and 2 to get 64.

$$-64\sqrt{5}+64+32\sqrt{10}+64\sqrt{5}-32\sqrt{10}$$

Combine -64\sqrt{5} and 64\sqrt{5} to get 0.

$$64+32\sqrt{10}-32\sqrt{10}$$

Combine 32\sqrt{10} and -32\sqrt{10} to get 0.

$$64$$

Factor

$$2^{6}$$