Solve (a^2)(4a^5)(-2a^3)

$$( a ^ { 2 } ) ( 4 a ^ { 5 } ) ( - 2 a ^ { 3 } )$$

Evaluate

$$-8a^{10}$$

Solution Steps

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

$$a^{7}\times 4\left(-2\right)a^{3}$$

To multiply powers of the same base, add their exponents. Add 7 and 3 to get 10.

$$a^{10}\times 4\left(-2\right)$$

Multiply 4 and -2 to get -8.

$$a^{10}\left(-8\right)$$

Differentiate w.r.t. a

$$-80a^{9}$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{7}\times 4\left(-2\right)a^{3})$$

To multiply powers of the same base, add their exponents. Add 7 and 3 to get 10.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{10}\times 4\left(-2\right))$$

Multiply 4 and -2 to get -8.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{10}\left(-8\right))$$

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

$$10\left(-8\right)a^{10-1}$$

Multiply 10 times -8.

$$-80a^{10-1}$$

Subtract 1 from 10.

$$-80a^{9}$$