Solve frac{(2z^frac{1{5}})^4}{z^frac{1{20}}}

$$\frac { ( 2 z ^ { \frac { 1 } { 5 } } ) ^ { 4 } } { z ^ { \frac { 1 } { 20 } } }$$

Evaluate

$$16z^{\frac{3}{4}}$$

Solution Steps

Expand \left(2z^{\frac{1}{5}}\right)^{4}.

$$\frac{2^{4}\left(z^{\frac{1}{5}}\right)^{4}}{z^{\frac{1}{20}}}$$

To raise a power to another power, multiply the exponents. Multiply \frac{1}{5} and 4 to get \frac{4}{5}.

$$\frac{2^{4}z^{\frac{4}{5}}}{z^{\frac{1}{20}}}$$

Calculate 2 to the power of 4 and get 16.

$$\frac{16z^{\frac{4}{5}}}{z^{\frac{1}{20}}}$$

Cancel out \sqrt[20]{z} in both numerator and denominator.

$$16z^{\frac{3}{4}}$$

Differentiate w.r.t. z

$$\frac{12}{\sqrt[4]{z}}$$

Steps Using Definition of a Derivative

Expand \left(2z^{\frac{1}{5}}\right)^{4}.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{2^{4}\left(z^{\frac{1}{5}}\right)^{4}}{z^{\frac{1}{20}}})$$

To raise a power to another power, multiply the exponents. Multiply \frac{1}{5} and 4 to get \frac{4}{5}.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{2^{4}z^{\frac{4}{5}}}{z^{\frac{1}{20}}})$$

Calculate 2 to the power of 4 and get 16.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{16z^{\frac{4}{5}}}{z^{\frac{1}{20}}})$$

Cancel out \sqrt[20]{z} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}z}(16z^{\frac{3}{4}})$$

The derivative of ax^{n} is nax^{n-1}.

$$\frac{3}{4}\times 16z^{\frac{3}{4}-1}$$

Multiply \frac{3}{4} times 16.

$$12z^{\frac{3}{4}-1}$$

Subtract 1 from \frac{3}{4}.

$$12z^{-\frac{1}{4}}$$