Solve frac{2x^{-3}y^-3}{(x^2y^4)^-frac{1{6}}}

$$\frac { 2 x ^ { - 3 } y ^ { - 3 } } { ( x ^ { 2 } y ^ { 4 } ) ^ { - \frac { 1 } { 6 } } }$$

Evaluate

$$\frac{2\sqrt[3]{|x|}}{y^{\frac{7}{3}}x^{3}}$$

Solution Steps

Expand \left(x^{2}y^{4}\right)^{-\frac{1}{6}}.

$$\frac{2x^{-3}y^{-3}}{\left(x^{2}\right)^{-\frac{1}{6}}\left(y^{4}\right)^{-\frac{1}{6}}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and -\frac{1}{6} to get -\frac{1}{3}.

$$\frac{2x^{-3}y^{-3}}{x^{-\frac{1}{3}}\left(y^{4}\right)^{-\frac{1}{6}}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and -\frac{1}{6} to get -\frac{2}{3}.

$$\frac{2x^{-3}y^{-3}}{x^{-\frac{1}{3}}y^{-\frac{2}{3}}}$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{2}{y^{\frac{7}{3}}x^{\frac{8}{3}}}$$

Differentiate w.r.t. x

$$-\frac{16\sqrt[3]{|x|}}{3y^{\frac{7}{3}}x^{4}}$$