Solve 628sqrt{9/10+frac{785^2/984^2}}{9/10+785^2/984^2}

$$\frac { 628 \sqrt { \frac { 9 } { 10 } + \frac { 785 ^ { 2 } } { 984 ^ { 2 } } } } { \frac { 9 } { 10 } + \frac { 785 ^ { 2 } } { 984 ^ { 2 } } }$$

Evaluate

$$\frac{617952\sqrt{37191385}}{7438277}\approx 506.644777914$$

Solution Steps

Calculate 785 to the power of 2 and get 616225.

$$\frac{628\sqrt{\frac{9}{10}+\frac{616225}{984^{2}}}}{\frac{9}{10}+\frac{785^{2}}{984^{2}}}$$

Calculate 984 to the power of 2 and get 968256.

$$\frac{628\sqrt{\frac{9}{10}+\frac{616225}{968256}}}{\frac{9}{10}+\frac{785^{2}}{984^{2}}}$$

Least common multiple of 10 and 968256 is 4841280. Convert \frac{9}{10} and \frac{616225}{968256} to fractions with denominator 4841280.

$$\frac{628\sqrt{\frac{4357152}{4841280}+\frac{3081125}{4841280}}}{\frac{9}{10}+\frac{785^{2}}{984^{2}}}$$

Since \frac{4357152}{4841280} and \frac{3081125}{4841280} have the same denominator, add them by adding their numerators.

$$\frac{628\sqrt{\frac{4357152+3081125}{4841280}}}{\frac{9}{10}+\frac{785^{2}}{984^{2}}}$$

Add 4357152 and 3081125 to get 7438277.

$$\frac{628\sqrt{\frac{7438277}{4841280}}}{\frac{9}{10}+\frac{785^{2}}{984^{2}}}$$

Rewrite the square root of the division \sqrt{\frac{7438277}{4841280}} as the division of square roots \frac{\sqrt{7438277}}{\sqrt{4841280}}.

$$\frac{628\times \frac{\sqrt{7438277}}{\sqrt{4841280}}}{\frac{9}{10}+\frac{785^{2}}{984^{2}}}$$

Factor 4841280=984^{2}\times 5. Rewrite the square root of the product \sqrt{984^{2}\times 5} as the product of square roots \sqrt{984^{2}}\sqrt{5}. Take the square root of 984^{2}.

$$\frac{628\times \frac{\sqrt{7438277}}{984\sqrt{5}}}{\frac{9}{10}+\frac{785^{2}}{984^{2}}}$$

Rationalize the denominator of \frac{\sqrt{7438277}}{984\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{628\times \frac{\sqrt{7438277}\sqrt{5}}{984\left(\sqrt{5}\right)^{2}}}{\frac{9}{10}+\frac{785^{2}}{984^{2}}}$$

The square of \sqrt{5} is 5.

$$\frac{628\times \frac{\sqrt{7438277}\sqrt{5}}{984\times 5}}{\frac{9}{10}+\frac{785^{2}}{984^{2}}}$$

To multiply \sqrt{7438277} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{628\times \frac{\sqrt{37191385}}{984\times 5}}{\frac{9}{10}+\frac{785^{2}}{984^{2}}}$$

Multiply 984 and 5 to get 4920.

$$\frac{628\times \frac{\sqrt{37191385}}{4920}}{\frac{9}{10}+\frac{785^{2}}{984^{2}}}$$

Express 628\times \frac{\sqrt{37191385}}{4920} as a single fraction.

$$\frac{\frac{628\sqrt{37191385}}{4920}}{\frac{9}{10}+\frac{785^{2}}{984^{2}}}$$

Calculate 785 to the power of 2 and get 616225.

$$\frac{\frac{628\sqrt{37191385}}{4920}}{\frac{9}{10}+\frac{616225}{984^{2}}}$$

Calculate 984 to the power of 2 and get 968256.

$$\frac{\frac{628\sqrt{37191385}}{4920}}{\frac{9}{10}+\frac{616225}{968256}}$$

Least common multiple of 10 and 968256 is 4841280. Convert \frac{9}{10} and \frac{616225}{968256} to fractions with denominator 4841280.

$$\frac{\frac{628\sqrt{37191385}}{4920}}{\frac{4357152}{4841280}+\frac{3081125}{4841280}}$$

Since \frac{4357152}{4841280} and \frac{3081125}{4841280} have the same denominator, add them by adding their numerators.

$$\frac{\frac{628\sqrt{37191385}}{4920}}{\frac{4357152+3081125}{4841280}}$$

Add 4357152 and 3081125 to get 7438277.

$$\frac{\frac{628\sqrt{37191385}}{4920}}{\frac{7438277}{4841280}}$$

Divide \frac{628\sqrt{37191385}}{4920} by \frac{7438277}{4841280} by multiplying \frac{628\sqrt{37191385}}{4920} by the reciprocal of \frac{7438277}{4841280}.

$$\frac{628\sqrt{37191385}\times 4841280}{4920\times 7438277}$$

Cancel out 4\times 1230 in both numerator and denominator.

$$\frac{157\times 3936\sqrt{37191385}}{7438277}$$

Multiply 157 and 3936 to get 617952.

$$\frac{617952\sqrt{37191385}}{7438277}$$