Solve (frac{y^{6}z^-3}{x^-1t^3})^frac{1{3}}

$$( \frac { y ^ { 6 } z ^ { - 3 } } { x ^ { - 1 } t ^ { 3 } } ) ^ { \frac { 1 } { 3 } }$$

Evaluate

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

Solution Steps

To raise \frac{y^{6}z^{-3}}{x^{-1}t^{3}} to a power, raise both numerator and denominator to the power and then divide.

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

Expand \left(y^{6}z^{-3}\right)^{\frac{1}{3}}.

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

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

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

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

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

Expand \left(x^{-1}t^{3}\right)^{\frac{1}{3}}.

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

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

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

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

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

Calculate t to the power of 1 and get t.

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

Differentiate w.r.t. x

$$\frac{y^{2}}{3x^{\frac{2}{3}}tz}$$