Solve (81/16)^{0,25}*((-8)^-frac{4{3}}*a^-8)*(2^-2*a^30)=

$$( \frac { 81 } { 16 } ) ^ { 0,25 } \cdot ( ( - 8 ) ^ { - \frac { 4 } { 3 } } \cdot a ^ { - 8 } ) \cdot ( 2 ^ { - 2 } \cdot a ^ { 30 } ) =$$

Evaluate

$$\frac{3a^{22}}{128}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add -8 and 30 to get 22.

$$\left(\frac{81}{16}\right)^{0,25}\left(-8\right)^{-\frac{4}{3}}a^{22}\times 2^{-2}$$

Calculate \frac{81}{16} to the power of 0,25 and get \frac{3}{2}.

$$\frac{3}{2}\left(-8\right)^{-\frac{4}{3}}a^{22}\times 2^{-2}$$

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

$$\frac{3}{2}\times \frac{1}{16}a^{22}\times 2^{-2}$$

Multiply \frac{3}{2} and \frac{1}{16} to get \frac{3}{32}.

$$\frac{3}{32}a^{22}\times 2^{-2}$$

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

$$\frac{3}{32}a^{22}\times \frac{1}{4}$$

Multiply \frac{3}{32} and \frac{1}{4} to get \frac{3}{128}.

$$\frac{3}{128}a^{22}$$

Differentiate w.r.t. a

$$\frac{33a^{21}}{64}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add -8 and 30 to get 22.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\left(\frac{81}{16}\right)^{0,25}\left(-8\right)^{-\frac{4}{3}}a^{22}\times 2^{-2})$$

Calculate \frac{81}{16} to the power of 0,25 and get \frac{3}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{3}{2}\left(-8\right)^{-\frac{4}{3}}a^{22}\times 2^{-2})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{3}{2}\times \frac{1}{16}a^{22}\times 2^{-2})$$

Multiply \frac{3}{2} and \frac{1}{16} to get \frac{3}{32}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{3}{32}a^{22}\times 2^{-2})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{3}{32}a^{22}\times \frac{1}{4})$$

Multiply \frac{3}{32} and \frac{1}{4} to get \frac{3}{128}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{3}{128}a^{22})$$

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

$$22\times \frac{3}{128}a^{22-1}$$

Multiply 22 times \frac{3}{128}.

$$\frac{33}{64}a^{22-1}$$

Subtract 1 from 22.

$$\frac{33}{64}a^{21}$$