Solve sqrt{76.79-31^2/15/14}

$$\sqrt{ \frac{ 76.79- \frac{ { 31 }^{ 2 } }{ 15 } }{ 14 } }$$

Evaluate

$$\frac{\sqrt{160314}}{420}\approx 0.953315018$$

Solution Steps

Calculate 31 to the power of 2 and get 961.

$$\sqrt{\frac{76.79-\frac{961}{15}}{14}}$$

Convert decimal number 76.79 to fraction \frac{7679}{100}.

$$\sqrt{\frac{\frac{7679}{100}-\frac{961}{15}}{14}}$$

Least common multiple of 100 and 15 is 300. Convert \frac{7679}{100} and \frac{961}{15} to fractions with denominator 300.

$$\sqrt{\frac{\frac{23037}{300}-\frac{19220}{300}}{14}}$$

Since \frac{23037}{300} and \frac{19220}{300} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{\frac{23037-19220}{300}}{14}}$$

Subtract 19220 from 23037 to get 3817.

$$\sqrt{\frac{\frac{3817}{300}}{14}}$$

Express \frac{\frac{3817}{300}}{14} as a single fraction.

$$\sqrt{\frac{3817}{300\times 14}}$$

Multiply 300 and 14 to get 4200.

$$\sqrt{\frac{3817}{4200}}$$

Rewrite the square root of the division \sqrt{\frac{3817}{4200}} as the division of square roots \frac{\sqrt{3817}}{\sqrt{4200}}.

$$\frac{\sqrt{3817}}{\sqrt{4200}}$$

Factor 4200=10^{2}\times 42. Rewrite the square root of the product \sqrt{10^{2}\times 42} as the product of square roots \sqrt{10^{2}}\sqrt{42}. Take the square root of 10^{2}.

$$\frac{\sqrt{3817}}{10\sqrt{42}}$$

Rationalize the denominator of \frac{\sqrt{3817}}{10\sqrt{42}} by multiplying numerator and denominator by \sqrt{42}.

$$\frac{\sqrt{3817}\sqrt{42}}{10\left(\sqrt{42}\right)^{2}}$$

The square of \sqrt{42} is 42.

$$\frac{\sqrt{3817}\sqrt{42}}{10\times 42}$$

To multiply \sqrt{3817} and \sqrt{42}, multiply the numbers under the square root.

$$\frac{\sqrt{160314}}{10\times 42}$$

Multiply 10 and 42 to get 420.

$$\frac{\sqrt{160314}}{420}$$