Solve 5/x^3sqrt{91493*525div81/8}

$$\frac{ 5 }{ { x }^{ 3 } } \sqrt{ 91493 \times 525 \div \frac{ 81 }{ 8 } }$$

Evaluate

$$\frac{50\sqrt{3842706}}{9x^{3}}$$

Solution Steps

Multiply 91493 and 525 to get 48033825.

$$\frac{5}{x^{3}}\sqrt{\frac{48033825}{\frac{81}{8}}}$$

Divide 48033825 by \frac{81}{8} by multiplying 48033825 by the reciprocal of \frac{81}{8}.

$$\frac{5}{x^{3}}\sqrt{48033825\times \frac{8}{81}}$$

Multiply 48033825 and \frac{8}{81} to get \frac{128090200}{27}.

$$\frac{5}{x^{3}}\sqrt{\frac{128090200}{27}}$$

Rewrite the square root of the division \sqrt{\frac{128090200}{27}} as the division of square roots \frac{\sqrt{128090200}}{\sqrt{27}}.

$$\frac{5}{x^{3}}\times \frac{\sqrt{128090200}}{\sqrt{27}}$$

Factor 128090200=10^{2}\times 1280902. Rewrite the square root of the product \sqrt{10^{2}\times 1280902} as the product of square roots \sqrt{10^{2}}\sqrt{1280902}. Take the square root of 10^{2}.

$$\frac{5}{x^{3}}\times \frac{10\sqrt{1280902}}{\sqrt{27}}$$

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

$$\frac{5}{x^{3}}\times \frac{10\sqrt{1280902}}{3\sqrt{3}}$$

Rationalize the denominator of \frac{10\sqrt{1280902}}{3\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{5}{x^{3}}\times \frac{10\sqrt{1280902}\sqrt{3}}{3\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{5}{x^{3}}\times \frac{10\sqrt{1280902}\sqrt{3}}{3\times 3}$$

To multiply \sqrt{1280902} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{5}{x^{3}}\times \frac{10\sqrt{3842706}}{3\times 3}$$

Multiply 3 and 3 to get 9.

$$\frac{5}{x^{3}}\times \frac{10\sqrt{3842706}}{9}$$

Multiply \frac{5}{x^{3}} times \frac{10\sqrt{3842706}}{9} by multiplying numerator times numerator and denominator times denominator.

$$\frac{5\times 10\sqrt{3842706}}{x^{3}\times 9}$$

Multiply 5 and 10 to get 50.

$$\frac{50\sqrt{3842706}}{x^{3}\times 9}$$

Differentiate w.r.t. x

$$-\frac{50\sqrt{3842706}}{3x^{4}}$$