Solve (2x^2y^3)(-5x^-2y^7)

$$( 2 x ^ { 2 } y ^ { 3 } ) ( - 5 x ^ { - 2 } y ^ { 7 } )$$

Evaluate

$$-10y^{10}$$

Steps Using Commutative Property of Multiplication

Use the rules of exponents to simplify the expression.

$$2^{1}x^{2}y^{3}\left(-5\right)^{1}x^{-2}y^{7}$$

Use the Commutative Property of Multiplication.

$$2^{1}\left(-5\right)^{1}x^{2}x^{-2}y^{3}y^{7}$$

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

$$2^{1}\left(-5\right)^{1}x^{2-2}y^{3+7}$$

Add the exponents 2 and -2.

$$2^{1}\left(-5\right)^{1}x^{0}y^{3+7}$$

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

$$2^{1}\left(-5\right)^{1}y^{3+7}$$

Add the exponents 3 and 7.

$$2^{1}\left(-5\right)^{1}y^{10}$$

Multiply 2 times -5.

$$-10y^{10}$$

Differentiate w.r.t. y

$$-100y^{9}$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}y}(2y^{3}\left(-5\right)y^{7})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}y}(2y^{10}\left(-5\right))$$

Multiply 2 and -5 to get -10.

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

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

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

Multiply 10 times -10.

$$-100y^{10-1}$$

Subtract 1 from 10.

$$-100y^{9}$$