Solve sqrt{3.986*10^14*(2/7178137-1/7028137)}

$$\sqrt{ 3.986 \times { 10 }^{ 14 } \times ( \frac{ 2 }{ 7178137 } - \frac{ 1 }{ 7028137 } ) }$$

Evaluate

$$\frac{200000\sqrt{3457801724115040847672645}}{50448930240769}\approx 7371.876948808$$

Solution Steps

Calculate 10 to the power of 14 and get 100000000000000.

$$\sqrt{3.986\times 100000000000000\left(\frac{2}{7178137}-\frac{1}{7028137}\right)}$$

Multiply 3.986 and 100000000000000 to get 398600000000000.

$$\sqrt{398600000000000\left(\frac{2}{7178137}-\frac{1}{7028137}\right)}$$

Least common multiple of 7178137 and 7028137 is 50448930240769. Convert \frac{2}{7178137} and \frac{1}{7028137} to fractions with denominator 50448930240769.

$$\sqrt{398600000000000\left(\frac{14056274}{50448930240769}-\frac{7178137}{50448930240769}\right)}$$

Since \frac{14056274}{50448930240769} and \frac{7178137}{50448930240769} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{398600000000000\times \frac{14056274-7178137}{50448930240769}}$$

Subtract 7178137 from 14056274 to get 6878137.

$$\sqrt{398600000000000\times \frac{6878137}{50448930240769}}$$

Express 398600000000000\times \frac{6878137}{50448930240769} as a single fraction.

$$\sqrt{\frac{398600000000000\times 6878137}{50448930240769}}$$

Multiply 398600000000000 and 6878137 to get 2741625408200000000000.

$$\sqrt{\frac{2741625408200000000000}{50448930240769}}$$

Rewrite the square root of the division \sqrt{\frac{2741625408200000000000}{50448930240769}} as the division of square roots \frac{\sqrt{2741625408200000000000}}{\sqrt{50448930240769}}.

$$\frac{\sqrt{2741625408200000000000}}{\sqrt{50448930240769}}$$

Factor 2741625408200000000000=200000^{2}\times 68540635205. Rewrite the square root of the product \sqrt{200000^{2}\times 68540635205} as the product of square roots \sqrt{200000^{2}}\sqrt{68540635205}. Take the square root of 200000^{2}.

$$\frac{200000\sqrt{68540635205}}{\sqrt{50448930240769}}$$

Rationalize the denominator of \frac{200000\sqrt{68540635205}}{\sqrt{50448930240769}} by multiplying numerator and denominator by \sqrt{50448930240769}.

$$\frac{200000\sqrt{68540635205}\sqrt{50448930240769}}{\left(\sqrt{50448930240769}\right)^{2}}$$

The square of \sqrt{50448930240769} is 50448930240769.

$$\frac{200000\sqrt{68540635205}\sqrt{50448930240769}}{50448930240769}$$

To multiply \sqrt{68540635205} and \sqrt{50448930240769}, multiply the numbers under the square root.

$$\frac{200000\sqrt{3457801724115040847672645}}{50448930240769}$$