Solve frac{(40sqrt{85})^2+400^2-(40sqrt{145})^2}{2*40sqrt{85}*400}

$$\frac { ( 40 \sqrt { 85 } ) ^ { 2 } + 400 ^ { 2 } - ( 40 \sqrt { 145 } ) ^ { 2 } } { 2 \times 40 \sqrt { 85 } \times 400 }$$

Evaluate

$$\frac{2\sqrt{85}}{85}\approx 0.216930458$$

Solution Steps

Expand \left(40\sqrt{85}\right)^{2}.

$$\frac{40^{2}\left(\sqrt{85}\right)^{2}+400^{2}-\left(40\sqrt{145}\right)^{2}}{2\times 40\times 400\sqrt{85}}$$

Calculate 40 to the power of 2 and get 1600.

$$\frac{1600\left(\sqrt{85}\right)^{2}+400^{2}-\left(40\sqrt{145}\right)^{2}}{2\times 40\times 400\sqrt{85}}$$

The square of \sqrt{85} is 85.

$$\frac{1600\times 85+400^{2}-\left(40\sqrt{145}\right)^{2}}{2\times 40\times 400\sqrt{85}}$$

Multiply 1600 and 85 to get 136000.

$$\frac{136000+400^{2}-\left(40\sqrt{145}\right)^{2}}{2\times 40\times 400\sqrt{85}}$$

Calculate 400 to the power of 2 and get 160000.

$$\frac{136000+160000-\left(40\sqrt{145}\right)^{2}}{2\times 40\times 400\sqrt{85}}$$

Add 136000 and 160000 to get 296000.

$$\frac{296000-\left(40\sqrt{145}\right)^{2}}{2\times 40\times 400\sqrt{85}}$$

Expand \left(40\sqrt{145}\right)^{2}.

$$\frac{296000-40^{2}\left(\sqrt{145}\right)^{2}}{2\times 40\times 400\sqrt{85}}$$

Calculate 40 to the power of 2 and get 1600.

$$\frac{296000-1600\left(\sqrt{145}\right)^{2}}{2\times 40\times 400\sqrt{85}}$$

The square of \sqrt{145} is 145.

$$\frac{296000-1600\times 145}{2\times 40\times 400\sqrt{85}}$$

Multiply 1600 and 145 to get 232000.

$$\frac{296000-232000}{2\times 40\times 400\sqrt{85}}$$

Subtract 232000 from 296000 to get 64000.

$$\frac{64000}{2\times 40\times 400\sqrt{85}}$$

Multiply 2 and 40 to get 80.

$$\frac{64000}{80\times 400\sqrt{85}}$$

Multiply 80 and 400 to get 32000.

$$\frac{64000}{32000\sqrt{85}}$$

Rationalize the denominator of \frac{64000}{32000\sqrt{85}} by multiplying numerator and denominator by \sqrt{85}.

$$\frac{64000\sqrt{85}}{32000\left(\sqrt{85}\right)^{2}}$$

The square of \sqrt{85} is 85.

$$\frac{64000\sqrt{85}}{32000\times 85}$$

Cancel out 32000 in both numerator and denominator.

$$\frac{2\sqrt{85}}{85}$$