Solve a^-3*8^-7*8^5

$$a ^ { - 3 } \cdot 8 ^ { - 7 } \cdot 8 ^ { 5 }$$

Evaluate

$$\frac{1}{64a^{3}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add -7 and 5 to get -2.

$$a^{-3}\times 8^{-2}$$

Calculate 8 to the power of -2 and get \frac{1}{64}.

$$a^{-3}\times \frac{1}{64}$$

Differentiate w.r.t. a

$$-\frac{3}{64a^{4}}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add -7 and 5 to get -2.

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

Calculate 8 to the power of -2 and get \frac{1}{64}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{-3}\times \frac{1}{64})$$

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

$$-3\times \frac{1}{64}a^{-3-1}$$

Multiply -3 times \frac{1}{64}.

$$-\frac{3}{64}a^{-3-1}$$

Subtract 1 from -3.

$$-\frac{3}{64}a^{-4}$$