Solve sqrt{1/50}lfloor44494.25-{left(1345/50right)}^2rfloor

$$\sqrt{ \frac{ 1 }{ 50 } } \lfloor 44494.25- { \left( \frac{ 1345 }{ 50 } \right) }^{ 2 } \rfloor$$

Evaluate

$$4377\sqrt{2}\approx 6190.012762507$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{1}{50}} as the division of square roots \frac{\sqrt{1}}{\sqrt{50}}.

$$\frac{\sqrt{1}}{\sqrt{50}}\lfloor 44494.25-\left(\frac{1345}{50}\right)^{2}\rfloor$$

Calculate the square root of 1 and get 1.

$$\frac{1}{\sqrt{50}}\lfloor 44494.25-\left(\frac{1345}{50}\right)^{2}\rfloor$$

Factor 50=5^{2}\times 2. Rewrite the square root of the product \sqrt{5^{2}\times 2} as the product of square roots \sqrt{5^{2}}\sqrt{2}. Take the square root of 5^{2}.

$$\frac{1}{5\sqrt{2}}\lfloor 44494.25-\left(\frac{1345}{50}\right)^{2}\rfloor$$

Rationalize the denominator of \frac{1}{5\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\sqrt{2}}{5\left(\sqrt{2}\right)^{2}}\lfloor 44494.25-\left(\frac{1345}{50}\right)^{2}\rfloor$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{2}}{5\times 2}\lfloor 44494.25-\left(\frac{1345}{50}\right)^{2}\rfloor$$

Multiply 5 and 2 to get 10.

$$\frac{\sqrt{2}}{10}\lfloor 44494.25-\left(\frac{1345}{50}\right)^{2}\rfloor$$

Reduce the fraction \frac{1345}{50} to lowest terms by extracting and canceling out 5.

$$\frac{\sqrt{2}}{10}\lfloor 44494.25-\left(\frac{269}{10}\right)^{2}\rfloor$$

Calculate \frac{269}{10} to the power of 2 and get \frac{72361}{100}.

$$\frac{\sqrt{2}}{10}\lfloor 44494.25-\frac{72361}{100}\rfloor$$

Subtract \frac{72361}{100} from 44494.25 to get \frac{1094266}{25}.

$$\frac{\sqrt{2}}{10}\lfloor \frac{1094266}{25}\rfloor$$

Dividing 1094266 by 25 gives 43770 and remainder 16. Rewrite \frac{1094266}{25} as 43770+\frac{16}{25}.

$$\frac{\sqrt{2}}{10}\lfloor 43770+\frac{16}{25}\rfloor$$

The floor of a real number a is the largest integer number less than or equal to a. The floor of 43770+\frac{16}{25} is 43770.

$$\frac{\sqrt{2}}{10}\times 43770$$

Cancel out 10, the greatest common factor in 43770 and 10.

$$4377\sqrt{2}$$