Solve frac{(80sqrt{2}-16sqrt{10})(20sqrt{2}-4sqrt{10)}}{2}

$$\frac{ (80 \sqrt{ 2 } -16 \sqrt{ 10 } )(20 \sqrt{ 2 } -4 \sqrt{ 10) } }{ 2 }$$

Evaluate

$$1920-640\sqrt{5}\approx 488.9164944$$

Solution Steps

Apply the distributive property by multiplying each term of 80\sqrt{2}-16\sqrt{10} by each term of 20\sqrt{2}-4\sqrt{10}.

$$\frac{1600\left(\sqrt{2}\right)^{2}-320\sqrt{10}\sqrt{2}-320\sqrt{10}\sqrt{2}+64\left(\sqrt{10}\right)^{2}}{2}$$

The square of \sqrt{2} is 2.

$$\frac{1600\times 2-320\sqrt{10}\sqrt{2}-320\sqrt{10}\sqrt{2}+64\left(\sqrt{10}\right)^{2}}{2}$$

Multiply 1600 and 2 to get 3200.

$$\frac{3200-320\sqrt{10}\sqrt{2}-320\sqrt{10}\sqrt{2}+64\left(\sqrt{10}\right)^{2}}{2}$$

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}.

$$\frac{3200-320\sqrt{2}\sqrt{5}\sqrt{2}-320\sqrt{10}\sqrt{2}+64\left(\sqrt{10}\right)^{2}}{2}$$

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

$$\frac{3200-320\times 2\sqrt{5}-320\sqrt{10}\sqrt{2}+64\left(\sqrt{10}\right)^{2}}{2}$$

Multiply -320 and 2 to get -640.

$$\frac{3200-640\sqrt{5}-320\sqrt{10}\sqrt{2}+64\left(\sqrt{10}\right)^{2}}{2}$$

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}.

$$\frac{3200-640\sqrt{5}-320\sqrt{2}\sqrt{5}\sqrt{2}+64\left(\sqrt{10}\right)^{2}}{2}$$

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

$$\frac{3200-640\sqrt{5}-320\times 2\sqrt{5}+64\left(\sqrt{10}\right)^{2}}{2}$$

Multiply -320 and 2 to get -640.

$$\frac{3200-640\sqrt{5}-640\sqrt{5}+64\left(\sqrt{10}\right)^{2}}{2}$$

Combine -640\sqrt{5} and -640\sqrt{5} to get -1280\sqrt{5}.

$$\frac{3200-1280\sqrt{5}+64\left(\sqrt{10}\right)^{2}}{2}$$

The square of \sqrt{10} is 10.

$$\frac{3200-1280\sqrt{5}+64\times 10}{2}$$

Multiply 64 and 10 to get 640.

$$\frac{3200-1280\sqrt{5}+640}{2}$$

Add 3200 and 640 to get 3840.

$$\frac{3840-1280\sqrt{5}}{2}$$

Factor

$$640 {(3 - \sqrt{5})} = 488.9164944$$