Solve a^9*a^-4*a^7*a^-15

$$a ^ { 9 } \cdot a ^ { - 4 } \cdot a ^ { 7 } \cdot a ^ { - 15 }$$

Evaluate

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

Solution Steps

To multiply powers of the same base, add their exponents. Add 9 and -4 to get 5.

$$a^{5}a^{7}a^{-15}$$

To multiply powers of the same base, add their exponents. Add 5 and 7 to get 12.

$$a^{12}a^{-15}$$

To multiply powers of the same base, add their exponents. Add 12 and -15 to get -3.

$$a^{-3}$$

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 9 and -4 to get 5.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{5}a^{7}a^{-15})$$

To multiply powers of the same base, add their exponents. Add 5 and 7 to get 12.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{12}a^{-15})$$

To multiply powers of the same base, add their exponents. Add 12 and -15 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.

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