Solve sqrt{9-1910.045^2/199}

$$\sqrt{ \frac{ 9-191 { 0.045 }^{ 2 } }{ 199 } }$$

Evaluate

$$\frac{3\sqrt{7617919}}{39800}\approx 0.208044557$$

Solution Steps

Calculate 0.045 to the power of 2 and get 0.002025.

$$\sqrt{\frac{9-191\times 0.002025}{199}}$$

Multiply 191 and 0.002025 to get 0.386775.

$$\sqrt{\frac{9-0.386775}{199}}$$

Subtract 0.386775 from 9 to get 8.613225.

$$\sqrt{\frac{8.613225}{199}}$$

Expand \frac{8.613225}{199} by multiplying both numerator and the denominator by 1000000.

$$\sqrt{\frac{8613225}{199000000}}$$

Reduce the fraction \frac{8613225}{199000000} to lowest terms by extracting and canceling out 25.

$$\sqrt{\frac{344529}{7960000}}$$

Rewrite the square root of the division \sqrt{\frac{344529}{7960000}} as the division of square roots \frac{\sqrt{344529}}{\sqrt{7960000}}.

$$\frac{\sqrt{344529}}{\sqrt{7960000}}$$

Factor 344529=3^{2}\times 38281. Rewrite the square root of the product \sqrt{3^{2}\times 38281} as the product of square roots \sqrt{3^{2}}\sqrt{38281}. Take the square root of 3^{2}.

$$\frac{3\sqrt{38281}}{\sqrt{7960000}}$$

Factor 7960000=200^{2}\times 199. Rewrite the square root of the product \sqrt{200^{2}\times 199} as the product of square roots \sqrt{200^{2}}\sqrt{199}. Take the square root of 200^{2}.

$$\frac{3\sqrt{38281}}{200\sqrt{199}}$$

Rationalize the denominator of \frac{3\sqrt{38281}}{200\sqrt{199}} by multiplying numerator and denominator by \sqrt{199}.

$$\frac{3\sqrt{38281}\sqrt{199}}{200\left(\sqrt{199}\right)^{2}}$$

The square of \sqrt{199} is 199.

$$\frac{3\sqrt{38281}\sqrt{199}}{200\times 199}$$

To multiply \sqrt{38281} and \sqrt{199}, multiply the numbers under the square root.

$$\frac{3\sqrt{7617919}}{200\times 199}$$

Multiply 200 and 199 to get 39800.

$$\frac{3\sqrt{7617919}}{39800}$$