Solve 78/78sqrt{78(7896)}+7852155^2*45sqrt{451/781}

$$\frac{ 78 }{ 78 \sqrt{ 78(7896) } } + { 7852155 }^{ 2 } \times 45 \sqrt{ \frac{ 451 }{ 781 } }$$

Evaluate

$$\frac{\sqrt{4277}}{51324}+\frac{2774535216481125\sqrt{2911}}{71}\approx 2.108399942 \cdot 10^{15}$$

Short Solution Steps

Multiply 78 and 7896 to get 615888.

$$\frac{78}{78\sqrt{615888}}+45\times 7852155^{2}\sqrt{\frac{451}{781}}$$

Factor 615888=12^{2}\times 4277. Rewrite the square root of the product \sqrt{12^{2}\times 4277} as the product of square roots \sqrt{12^{2}}\sqrt{4277}. Take the square root of 12^{2}.

$$\frac{78}{78\times 12\sqrt{4277}}+45\times 7852155^{2}\sqrt{\frac{451}{781}}$$

Multiply 78 and 12 to get 936.

$$\frac{78}{936\sqrt{4277}}+45\times 7852155^{2}\sqrt{\frac{451}{781}}$$

Rationalize the denominator of \frac{78}{936\sqrt{4277}} by multiplying numerator and denominator by \sqrt{4277}.

$$\frac{78\sqrt{4277}}{936\left(\sqrt{4277}\right)^{2}}+45\times 7852155^{2}\sqrt{\frac{451}{781}}$$

The square of \sqrt{4277} is 4277.

$$\frac{78\sqrt{4277}}{936\times 4277}+45\times 7852155^{2}\sqrt{\frac{451}{781}}$$

Cancel out 78 in both numerator and denominator.

$$\frac{\sqrt{4277}}{12\times 4277}+45\times 7852155^{2}\sqrt{\frac{451}{781}}$$

Multiply 12 and 4277 to get 51324.

$$\frac{\sqrt{4277}}{51324}+45\times 7852155^{2}\sqrt{\frac{451}{781}}$$

Calculate 7852155 to the power of 2 and get 61656338144025.

$$\frac{\sqrt{4277}}{51324}+45\times 61656338144025\sqrt{\frac{451}{781}}$$

Multiply 45 and 61656338144025 to get 2774535216481125.

$$\frac{\sqrt{4277}}{51324}+2774535216481125\sqrt{\frac{451}{781}}$$

Reduce the fraction \frac{451}{781} to lowest terms by extracting and canceling out 11.

$$\frac{\sqrt{4277}}{51324}+2774535216481125\sqrt{\frac{41}{71}}$$

Rewrite the square root of the division \sqrt{\frac{41}{71}} as the division of square roots \frac{\sqrt{41}}{\sqrt{71}}.

$$\frac{\sqrt{4277}}{51324}+2774535216481125\times \frac{\sqrt{41}}{\sqrt{71}}$$

Rationalize the denominator of \frac{\sqrt{41}}{\sqrt{71}} by multiplying numerator and denominator by \sqrt{71}.

$$\frac{\sqrt{4277}}{51324}+2774535216481125\times \frac{\sqrt{41}\sqrt{71}}{\left(\sqrt{71}\right)^{2}}$$

The square of \sqrt{71} is 71.

$$\frac{\sqrt{4277}}{51324}+2774535216481125\times \frac{\sqrt{41}\sqrt{71}}{71}$$

To multiply \sqrt{41} and \sqrt{71}, multiply the numbers under the square root.

$$\frac{\sqrt{4277}}{51324}+2774535216481125\times \frac{\sqrt{2911}}{71}$$

Express 2774535216481125\times \frac{\sqrt{2911}}{71} as a single fraction.

$$\frac{\sqrt{4277}}{51324}+\frac{2774535216481125\sqrt{2911}}{71}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 51324 and 71 is 3644004. Multiply \frac{\sqrt{4277}}{51324} times \frac{71}{71}. Multiply \frac{2774535216481125\sqrt{2911}}{71} times \frac{51324}{51324}.

$$\frac{71\sqrt{4277}}{3644004}+\frac{51324\times 2774535216481125\sqrt{2911}}{3644004}$$

Since \frac{71\sqrt{4277}}{3644004} and \frac{51324\times 2774535216481125\sqrt{2911}}{3644004} have the same denominator, add them by adding their numerators.

$$\frac{71\sqrt{4277}+51324\times 2774535216481125\sqrt{2911}}{3644004}$$

Do the multiplications in 71\sqrt{4277}+51324\times 2774535216481125\sqrt{2911}.

$$\frac{71\sqrt{4277}+142400245450677259500\sqrt{2911}}{3644004}$$