Solve 12x31sqrt{1341/11143}

$$12x31 \sqrt{ \frac{ 1341 }{ 11143 } }$$

Evaluate

$$\frac{36\sqrt{1660307}x_{31}}{11143}$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{1341}{11143}} as the division of square roots \frac{\sqrt{1341}}{\sqrt{11143}}.

$$12x_{31}\times \frac{\sqrt{1341}}{\sqrt{11143}}$$

Factor 1341=3^{2}\times 149. Rewrite the square root of the product \sqrt{3^{2}\times 149} as the product of square roots \sqrt{3^{2}}\sqrt{149}. Take the square root of 3^{2}.

$$12x_{31}\times \frac{3\sqrt{149}}{\sqrt{11143}}$$

Rationalize the denominator of \frac{3\sqrt{149}}{\sqrt{11143}} by multiplying numerator and denominator by \sqrt{11143}.

$$12x_{31}\times \frac{3\sqrt{149}\sqrt{11143}}{\left(\sqrt{11143}\right)^{2}}$$

The square of \sqrt{11143} is 11143.

$$12x_{31}\times \frac{3\sqrt{149}\sqrt{11143}}{11143}$$

To multiply \sqrt{149} and \sqrt{11143}, multiply the numbers under the square root.

$$12x_{31}\times \frac{3\sqrt{1660307}}{11143}$$

Express 12\times \frac{3\sqrt{1660307}}{11143} as a single fraction.

$$\frac{12\times 3\sqrt{1660307}}{11143}x_{31}$$

Multiply 12 and 3 to get 36.

$$\frac{36\sqrt{1660307}}{11143}x_{31}$$

Express \frac{36\sqrt{1660307}}{11143}x_{31} as a single fraction.

$$\frac{36\sqrt{1660307}x_{31}}{11143}$$

Differentiate w.r.t. x_31

$$\frac{36 \sqrt{1660307}}{11143} = 4.162886496072137$$