Solve sqrt{3.896*10^14}sqrt{2/6900000-1/7050000}

$$\sqrt{ 3.896 \times { 10 }^{ 14 } } \sqrt{ \frac{ 2 }{ 6900000 } - \frac{ 1 }{ 7050000 } }$$

Evaluate

$$\frac{8000\sqrt{1052894}}{1081}\approx 7593.75565082$$

Solution Steps

Calculate 10 to the power of 14 and get 100000000000000.

$$\sqrt{3.896\times 100000000000000}\sqrt{\frac{2}{6900000}-\frac{1}{7050000}}$$

Multiply 3.896 and 100000000000000 to get 389600000000000.

$$\sqrt{389600000000000}\sqrt{\frac{2}{6900000}-\frac{1}{7050000}}$$

Factor 389600000000000=400000^{2}\times 2435. Rewrite the square root of the product \sqrt{400000^{2}\times 2435} as the product of square roots \sqrt{400000^{2}}\sqrt{2435}. Take the square root of 400000^{2}.

$$400000\sqrt{2435}\sqrt{\frac{2}{6900000}-\frac{1}{7050000}}$$

Reduce the fraction \frac{2}{6900000} to lowest terms by extracting and canceling out 2.

$$400000\sqrt{2435}\sqrt{\frac{1}{3450000}-\frac{1}{7050000}}$$

Least common multiple of 3450000 and 7050000 is 162150000. Convert \frac{1}{3450000} and \frac{1}{7050000} to fractions with denominator 162150000.

$$400000\sqrt{2435}\sqrt{\frac{47}{162150000}-\frac{23}{162150000}}$$

Since \frac{47}{162150000} and \frac{23}{162150000} have the same denominator, subtract them by subtracting their numerators.

$$400000\sqrt{2435}\sqrt{\frac{47-23}{162150000}}$$

Subtract 23 from 47 to get 24.

$$400000\sqrt{2435}\sqrt{\frac{24}{162150000}}$$

Reduce the fraction \frac{24}{162150000} to lowest terms by extracting and canceling out 24.

$$400000\sqrt{2435}\sqrt{\frac{1}{6756250}}$$

Rewrite the square root of the division \sqrt{\frac{1}{6756250}} as the division of square roots \frac{\sqrt{1}}{\sqrt{6756250}}.

$$400000\sqrt{2435}\times \frac{\sqrt{1}}{\sqrt{6756250}}$$

Calculate the square root of 1 and get 1.

$$400000\sqrt{2435}\times \frac{1}{\sqrt{6756250}}$$

Factor 6756250=25^{2}\times 10810. Rewrite the square root of the product \sqrt{25^{2}\times 10810} as the product of square roots \sqrt{25^{2}}\sqrt{10810}. Take the square root of 25^{2}.

$$400000\sqrt{2435}\times \frac{1}{25\sqrt{10810}}$$

Rationalize the denominator of \frac{1}{25\sqrt{10810}} by multiplying numerator and denominator by \sqrt{10810}.

$$400000\sqrt{2435}\times \frac{\sqrt{10810}}{25\left(\sqrt{10810}\right)^{2}}$$

The square of \sqrt{10810} is 10810.

$$400000\sqrt{2435}\times \frac{\sqrt{10810}}{25\times 10810}$$

Multiply 25 and 10810 to get 270250.

$$400000\sqrt{2435}\times \frac{\sqrt{10810}}{270250}$$

Express 400000\times \frac{\sqrt{10810}}{270250} as a single fraction.

$$\frac{400000\sqrt{10810}}{270250}\sqrt{2435}$$

Divide 400000\sqrt{10810} by 270250 to get \frac{1600}{1081}\sqrt{10810}.

$$\frac{1600}{1081}\sqrt{10810}\sqrt{2435}$$

To multiply \sqrt{10810} and \sqrt{2435}, multiply the numbers under the square root.

$$\frac{1600}{1081}\sqrt{26322350}$$

Factor 26322350=5^{2}\times 1052894. Rewrite the square root of the product \sqrt{5^{2}\times 1052894} as the product of square roots \sqrt{5^{2}}\sqrt{1052894}. Take the square root of 5^{2}.

$$\frac{1600}{1081}\times 5\sqrt{1052894}$$

Express \frac{1600}{1081}\times 5 as a single fraction.

$$\frac{1600\times 5}{1081}\sqrt{1052894}$$

Multiply 1600 and 5 to get 8000.

$$\frac{8000}{1081}\sqrt{1052894}$$