Solve sqrt{200(399335)-8835^2/200(200-1)}

$$\sqrt{ \frac{ 200(399335)- { 8835 }^{ 2 } }{ 200(200-1) } }$$

Evaluate

$$\frac{\sqrt{28811618}}{796}\approx 6.743273216$$

Solution Steps

Multiply 200 and 399335 to get 79867000.

$$\sqrt{\frac{79867000-8835^{2}}{200\left(200-1\right)}}$$

Calculate 8835 to the power of 2 and get 78057225.

$$\sqrt{\frac{79867000-78057225}{200\left(200-1\right)}}$$

Subtract 78057225 from 79867000 to get 1809775.

$$\sqrt{\frac{1809775}{200\left(200-1\right)}}$$

Subtract 1 from 200 to get 199.

$$\sqrt{\frac{1809775}{200\times 199}}$$

Multiply 200 and 199 to get 39800.

$$\sqrt{\frac{1809775}{39800}}$$

Reduce the fraction \frac{1809775}{39800} to lowest terms by extracting and canceling out 25.

$$\sqrt{\frac{72391}{1592}}$$

Rewrite the square root of the division \sqrt{\frac{72391}{1592}} as the division of square roots \frac{\sqrt{72391}}{\sqrt{1592}}.

$$\frac{\sqrt{72391}}{\sqrt{1592}}$$

Factor 1592=2^{2}\times 398. Rewrite the square root of the product \sqrt{2^{2}\times 398} as the product of square roots \sqrt{2^{2}}\sqrt{398}. Take the square root of 2^{2}.

$$\frac{\sqrt{72391}}{2\sqrt{398}}$$

Rationalize the denominator of \frac{\sqrt{72391}}{2\sqrt{398}} by multiplying numerator and denominator by \sqrt{398}.

$$\frac{\sqrt{72391}\sqrt{398}}{2\left(\sqrt{398}\right)^{2}}$$

The square of \sqrt{398} is 398.

$$\frac{\sqrt{72391}\sqrt{398}}{2\times 398}$$

To multiply \sqrt{72391} and \sqrt{398}, multiply the numbers under the square root.

$$\frac{\sqrt{28811618}}{2\times 398}$$

Multiply 2 and 398 to get 796.

$$\frac{\sqrt{28811618}}{796}$$