Solve a^3*a*a^-7

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

Evaluate

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

Steps Using Product of Powers Property

Use the rules of exponents to simplify the expression.

$$a^{3}a^{1}a^{-7}$$

Use the Multiplication Rule for Exponents.

$$a^{3+1-7}$$

Add the exponents 3 and 1.

$$a^{4-7}$$

Add the exponents 4 and -7.

$$a^{-3}$$

Differentiate w.r.t. a

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

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add 3 and 1 to get 4.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{4}a^{-7})$$

To multiply powers of the same base, add their exponents. Add 4 and -7 to get -3.

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

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

$$-3a^{-3-1}$$

Subtract 1 from -3.

$$-3a^{-4}$$