Solve (16v/81w)^-frac{3{4}}

$$( \frac { 16 v } { 81 w } ) ^ { - \frac { 3 } { 4 } }$$

Evaluate

$$\frac{27}{8\times \left(\frac{v}{w}\right)^{\frac{3}{4}}}$$

Solution Steps

To raise \frac{16v}{81w} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(16v\right)^{-\frac{3}{4}}}{\left(81w\right)^{-\frac{3}{4}}}$$

Expand \left(16v\right)^{-\frac{3}{4}}.

$$\frac{16^{-\frac{3}{4}}v^{-\frac{3}{4}}}{\left(81w\right)^{-\frac{3}{4}}}$$

Calculate 16 to the power of -\frac{3}{4} and get \frac{1}{8}.

$$\frac{\frac{1}{8}v^{-\frac{3}{4}}}{\left(81w\right)^{-\frac{3}{4}}}$$

Expand \left(81w\right)^{-\frac{3}{4}}.

$$\frac{\frac{1}{8}v^{-\frac{3}{4}}}{81^{-\frac{3}{4}}w^{-\frac{3}{4}}}$$

Calculate 81 to the power of -\frac{3}{4} and get \frac{1}{27}.

$$\frac{\frac{1}{8}v^{-\frac{3}{4}}}{\frac{1}{27}w^{-\frac{3}{4}}}$$

Differentiate w.r.t. v

$$-\frac{81}{32\times \left(\frac{v}{w}\right)^{\frac{7}{4}}w}$$