Solve (5n)(4n^-8)(8n^3)

$$( 5 n ) ( 4 n ^ { - 8 } ) ( 8 n ^ { 3 } )$$

Evaluate

$$\frac{160}{n^{4}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 1 and -8 to get -7.

$$5n^{-7}\times 4\times 8n^{3}$$

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

$$5n^{-4}\times 4\times 8$$

Multiply 5 and 4 to get 20.

$$20n^{-4}\times 8$$

Multiply 20 and 8 to get 160.

$$160n^{-4}$$

Differentiate w.r.t. n

$$-\frac{640}{n^{5}}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add 1 and -8 to get -7.

$$\frac{\mathrm{d}}{\mathrm{d}n}(5n^{-7}\times 4\times 8n^{3})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}n}(5n^{-4}\times 4\times 8)$$

Multiply 5 and 4 to get 20.

$$\frac{\mathrm{d}}{\mathrm{d}n}(20n^{-4}\times 8)$$

Multiply 20 and 8 to get 160.

$$\frac{\mathrm{d}}{\mathrm{d}n}(160n^{-4})$$

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

$$-4\times 160n^{-4-1}$$

Multiply -4 times 160.

$$-640n^{-4-1}$$

Subtract 1 from -4.

$$-640n^{-5}$$