Solve sqrt{16-400/109}

$$\sqrt { 16 - \frac { 400 } { 109 } }$$

Evaluate

$$\frac{8\sqrt{2289}}{109}\approx 3.511449164$$

Solution Steps

Convert 16 to fraction \frac{1744}{109}.

$$\sqrt{\frac{1744}{109}-\frac{400}{109}}$$

Since \frac{1744}{109} and \frac{400}{109} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{1744-400}{109}}$$

Subtract 400 from 1744 to get 1344.

$$\sqrt{\frac{1344}{109}}$$

Rewrite the square root of the division \sqrt{\frac{1344}{109}} as the division of square roots \frac{\sqrt{1344}}{\sqrt{109}}.

$$\frac{\sqrt{1344}}{\sqrt{109}}$$

Factor 1344=8^{2}\times 21. Rewrite the square root of the product \sqrt{8^{2}\times 21} as the product of square roots \sqrt{8^{2}}\sqrt{21}. Take the square root of 8^{2}.

$$\frac{8\sqrt{21}}{\sqrt{109}}$$

Rationalize the denominator of \frac{8\sqrt{21}}{\sqrt{109}} by multiplying numerator and denominator by \sqrt{109}.

$$\frac{8\sqrt{21}\sqrt{109}}{\left(\sqrt{109}\right)^{2}}$$

The square of \sqrt{109} is 109.

$$\frac{8\sqrt{21}\sqrt{109}}{109}$$

To multiply \sqrt{21} and \sqrt{109}, multiply the numbers under the square root.

$$\frac{8\sqrt{2289}}{109}$$