Solve ((81w^2z^-frac{1{2}})^-1)^frac{1{4}}

$$( ( 81 w ^ { 2 } z ^ { - \frac { 1 } { 2 } } ) ^ { - 1 } ) ^ { \frac { 1 } { 4 } }$$

Evaluate

$$\frac{\sqrt[8]{z}}{3\sqrt{|w|}}$$

Solution Steps

Expand \left(81w^{2}z^{-\frac{1}{2}}\right)^{-1}.

$$\left(81^{-1}\left(w^{2}\right)^{-1}\left(z^{-\frac{1}{2}}\right)^{-1}\right)^{\frac{1}{4}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and -1 to get -2.

$$\left(81^{-1}w^{-2}\left(z^{-\frac{1}{2}}\right)^{-1}\right)^{\frac{1}{4}}$$

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

$$\left(81^{-1}w^{-2}z^{\frac{1}{2}}\right)^{\frac{1}{4}}$$

Calculate 81 to the power of -1 and get \frac{1}{81}.

$$\left(\frac{1}{81}w^{-2}z^{\frac{1}{2}}\right)^{\frac{1}{4}}$$

Expand \left(\frac{1}{81}w^{-2}z^{\frac{1}{2}}\right)^{\frac{1}{4}}.

$$\left(\frac{1}{81}\right)^{\frac{1}{4}}\left(w^{-2}\right)^{\frac{1}{4}}\left(z^{\frac{1}{2}}\right)^{\frac{1}{4}}$$

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

$$\left(\frac{1}{81}\right)^{\frac{1}{4}}w^{-\frac{1}{2}}\left(z^{\frac{1}{2}}\right)^{\frac{1}{4}}$$

To raise a power to another power, multiply the exponents. Multiply \frac{1}{2} and \frac{1}{4} to get \frac{1}{8}.

$$\left(\frac{1}{81}\right)^{\frac{1}{4}}w^{-\frac{1}{2}}z^{\frac{1}{8}}$$

Calculate \frac{1}{81} to the power of \frac{1}{4} and get \frac{1}{3}.

$$\frac{1}{3}w^{-\frac{1}{2}}z^{\frac{1}{8}}$$

Differentiate w.r.t. z

$$\frac{1}{24\sqrt{|w|}z^{\frac{7}{8}}}$$