Solve (81/16)^{025}*((-8)^-frac{4{3}}*a^-8)*(2^-2*a^30)=

$$( \frac { 81 } { 16 } ) ^ { 025 } \cdot ( ( - 8 ) ^ { - \frac { 4 } { 3 } } \cdot a ^ { - 8 } ) \cdot ( 2 ^ { - 2 } \cdot a ^ { 30 } ) =$$

Evaluate

$$\frac{a^{22}}{64}$$

Solution Steps

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

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

Multiply 0 and 25 to get 0.

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

Calculate \frac{81}{16} to the power of 0 and get 1.

$$1\left(-8\right)^{-\frac{4}{3}}a^{22}\times 2^{-2}$$

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

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

Multiply 1 and \frac{1}{16} to get \frac{1}{16}.

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

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

$$\frac{1}{16}a^{22}\times \frac{1}{4}$$

Multiply \frac{1}{16} and \frac{1}{4} to get \frac{1}{64}.

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

Differentiate w.r.t. a

$$\frac{11a^{21}}{32}$$

Steps Using Definition of a Derivative

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

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

Multiply 0 and 25 to get 0.

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

Calculate \frac{81}{16} to the power of 0 and get 1.

$$\frac{\mathrm{d}}{\mathrm{d}a}(1\left(-8\right)^{-\frac{4}{3}}a^{22}\times 2^{-2})$$

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

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

Multiply 1 and \frac{1}{16} to get \frac{1}{16}.

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

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{16}a^{22}\times \frac{1}{4})$$

Multiply \frac{1}{16} and \frac{1}{4} to get \frac{1}{64}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{64}a^{22})$$

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

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

Multiply 22 times \frac{1}{64}.

$$\frac{11}{32}a^{22-1}$$

Subtract 1 from 22.

$$\frac{11}{32}a^{21}$$