Solve (a^{-5}b^7)(a^-2b^-7c^0)=

$$( a ^ { - 5 } b ^ { 7 } ) ( a ^ { - 2 } b ^ { - 7 } c ^ { 0 } ) =$$

Evaluate

$$\frac{1}{a^{7}}$$

Solution Steps

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

$$a^{-7}b^{7}b^{-7}c^{0}$$

Multiply b^{7} and b^{-7} to get 1.

$$a^{-7}c^{0}$$

Calculate c to the power of 0 and get 1.

$$a^{-7}\times 1$$

For any term t, t\times 1=t and 1t=t.

$$a^{-7}$$

Differentiate w.r.t. a

$$-\frac{7}{a^{8}}$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{-7}b^{7}b^{-7}c^{0})$$

Multiply b^{7} and b^{-7} to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{-7}c^{0})$$

Calculate c to the power of 0 and get 1.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{-7}\times 1)$$

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

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

Subtract 1 from -7.

$$-7a^{-8}$$