Solve (2^-3*a^6)^frac{2{3}}

$$( 2 ^ { - 3 } \times a ^ { 6 } ) ^ { \frac { 2 } { 3 } }$$

Evaluate

$$\frac{a^{4}}{4}$$

Solution Steps

Calculate 2 to the power of -3 and get \frac{1}{8}.

$$\left(\frac{1}{8}a^{6}\right)^{\frac{2}{3}}$$

Expand \left(\frac{1}{8}a^{6}\right)^{\frac{2}{3}}.

$$\left(\frac{1}{8}\right)^{\frac{2}{3}}\left(a^{6}\right)^{\frac{2}{3}}$$

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

$$\left(\frac{1}{8}\right)^{\frac{2}{3}}a^{4}$$

Calculate \frac{1}{8} to the power of \frac{2}{3} and get \frac{1}{4}.

$$\frac{1}{4}a^{4}$$

Differentiate w.r.t. a

$$a^{3}$$

Steps Using Definition of a Derivative

Calculate 2 to the power of -3 and get \frac{1}{8}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\left(\frac{1}{8}a^{6}\right)^{\frac{2}{3}})$$

Expand \left(\frac{1}{8}a^{6}\right)^{\frac{2}{3}}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\left(\frac{1}{8}\right)^{\frac{2}{3}}\left(a^{6}\right)^{\frac{2}{3}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\left(\frac{1}{8}\right)^{\frac{2}{3}}a^{4})$$

Calculate \frac{1}{8} to the power of \frac{2}{3} and get \frac{1}{4}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{4}a^{4})$$

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

$$4\times \frac{1}{4}a^{4-1}$$

Multiply 4 times \frac{1}{4}.

$$a^{4-1}$$

Subtract 1 from 4.

$$a^{3}$$