Solve (5r^2s^-2)(-2r^5s^2)

$$( 5 r ^ { 2 } s ^ { - 2 } ) ( - 2 r ^ { 5 } s ^ { 2 } )$$

Evaluate

$$-10r^{7}$$

Steps Using Commutative Property of Multiplication

Use the rules of exponents to simplify the expression.

$$5^{1}r^{2}s^{-2}\left(-2\right)^{1}r^{5}s^{2}$$

Use the Commutative Property of Multiplication.

$$5^{1}\left(-2\right)^{1}r^{2}r^{5}s^{-2}s^{2}$$

To multiply powers of the same base, add their exponents.

$$5^{1}\left(-2\right)^{1}r^{2+5}s^{-2+2}$$

Add the exponents 2 and 5.

$$5^{1}\left(-2\right)^{1}r^{7}s^{-2+2}$$

Add the exponents -2 and 2.

$$5^{1}\left(-2\right)^{1}r^{7}s^{0}$$

For any number a except 0, a^{0}=1.

$$5^{1}\left(-2\right)^{1}r^{7}$$

Multiply 5 times -2.

$$-10r^{7}$$

Differentiate w.r.t. r

$$-70r^{6}$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}r}(5r^{7}s^{-2}\left(-2\right)s^{2})$$

Multiply s^{-2} and s^{2} to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}r}(5r^{7}\left(-2\right))$$

Multiply 5 and -2 to get -10.

$$\frac{\mathrm{d}}{\mathrm{d}r}(-10r^{7})$$

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

$$7\left(-10\right)r^{7-1}$$

Multiply 7 times -10.

$$-70r^{7-1}$$

Subtract 1 from 7.

$$-70r^{6}$$