Solve frac{7-8sqrt{3}}{x1/sqrt{2}}

$$\frac { 7 - 8 \sqrt { 3 } } { x \frac { 1 } { \sqrt { 2 } } }$$

Evaluate

$$\frac{\sqrt{2}\left(7-8\sqrt{3}\right)}{x}$$

Short Solution Steps

Rationalize the denominator of \frac{1}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{7-8\sqrt{3}}{x\times \frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}}$$

The square of \sqrt{2} is 2.

$$\frac{7-8\sqrt{3}}{x\times \frac{\sqrt{2}}{2}}$$

Express x\times \frac{\sqrt{2}}{2} as a single fraction.

$$\frac{7-8\sqrt{3}}{\frac{x\sqrt{2}}{2}}$$

Divide 7-8\sqrt{3} by \frac{x\sqrt{2}}{2} by multiplying 7-8\sqrt{3} by the reciprocal of \frac{x\sqrt{2}}{2}.

$$\frac{\left(7-8\sqrt{3}\right)\times 2}{x\sqrt{2}}$$

Factor the expressions that are not already factored.

$$\frac{2\left(-8\sqrt{3}+7\right)}{\sqrt{2}x}$$

Cancel out \sqrt{2} in both numerator and denominator.

$$\frac{\sqrt{2}\left(-8\sqrt{3}+7\right)}{x}$$

Expand the expression.

$$\frac{-8\sqrt{2}\sqrt{3}+7\sqrt{2}}{x}$$

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{-8\sqrt{6}+7\sqrt{2}}{x}$$

Differentiate w.r.t. x

$$-\frac{\sqrt{2}\left(7-8\sqrt{3}\right)}{x^{2}}$$