Solve frac{20^2+{left(20sqrt{2}right)}^2-400}{2*20*20sqrt{2}}

$$\frac{ { 20 }^{ 2 } + { \left(20 \sqrt{ 2 } \right) }^{ 2 } -400 }{ 2 \times 20 \times 20 \sqrt{ 2 } }$$

Evaluate

$$\frac{\sqrt{2}}{2}\approx 0.707106781$$

Solution Steps

Calculate 20 to the power of 2 and get 400.

$$\frac{400+\left(20\sqrt{2}\right)^{2}-400}{2\times 20\times 20\sqrt{2}}$$

Expand \left(20\sqrt{2}\right)^{2}.

$$\frac{400+20^{2}\left(\sqrt{2}\right)^{2}-400}{2\times 20\times 20\sqrt{2}}$$

Calculate 20 to the power of 2 and get 400.

$$\frac{400+400\left(\sqrt{2}\right)^{2}-400}{2\times 20\times 20\sqrt{2}}$$

The square of \sqrt{2} is 2.

$$\frac{400+400\times 2-400}{2\times 20\times 20\sqrt{2}}$$

Multiply 400 and 2 to get 800.

$$\frac{400+800-400}{2\times 20\times 20\sqrt{2}}$$

Add 400 and 800 to get 1200.

$$\frac{1200-400}{2\times 20\times 20\sqrt{2}}$$

Subtract 400 from 1200 to get 800.

$$\frac{800}{2\times 20\times 20\sqrt{2}}$$

Multiply 2 and 20 to get 40.

$$\frac{800}{40\times 20\sqrt{2}}$$

Multiply 40 and 20 to get 800.

$$\frac{800}{800\sqrt{2}}$$

Rationalize the denominator of \frac{800}{800\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{800\sqrt{2}}{800\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{800\sqrt{2}}{800\times 2}$$

Cancel out 800 in both numerator and denominator.

$$\frac{\sqrt{2}}{2}$$