Solve (5b^8b^-12)(-10b^3b^-12)

$$( 5 b ^ { 8 } b ^ { - 12 } ) ( - 10 b ^ { 3 } b ^ { - 12 } )$$

Evaluate

$$-\frac{50}{b^{13}}$$

Steps Using Commutative Property of Multiplication

Use the rules of exponents to simplify the expression.

$$5^{1}b^{8}b^{-12}\left(-10\right)^{1}b^{3}b^{-12}$$

Use the Commutative Property of Multiplication.

$$5^{1}\left(-10\right)^{1}b^{8}b^{3}b^{-12}b^{-12}$$

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

$$5^{1}\left(-10\right)^{1}b^{8+3}b^{-12-12}$$

Add the exponents 8 and 3.

$$5^{1}\left(-10\right)^{1}b^{11}b^{-12-12}$$

Add the exponents -12 and -12.

$$5^{1}\left(-10\right)^{1}b^{11}b^{-24}$$

Multiply 5 times -10.

$$-50b^{11}\times \frac{1}{b^{24}}$$

Differentiate w.r.t. b

$$\frac{650}{b^{14}}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add 8 and -12 to get -4.

$$\frac{\mathrm{d}}{\mathrm{d}b}(5b^{-4}\left(-10\right)b^{3}b^{-12})$$

To multiply powers of the same base, add their exponents. Add -4 and 3 to get -1.

$$\frac{\mathrm{d}}{\mathrm{d}b}(5b^{-1}\left(-10\right)b^{-12})$$

To multiply powers of the same base, add their exponents. Add -1 and -12 to get -13.

$$\frac{\mathrm{d}}{\mathrm{d}b}(5b^{-13}\left(-10\right))$$

Multiply 5 and -10 to get -50.

$$\frac{\mathrm{d}}{\mathrm{d}b}(-50b^{-13})$$

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

$$-13\left(-50\right)b^{-13-1}$$

Multiply -13 times -50.

$$650b^{-13-1}$$

Subtract 1 from -13.

$$650b^{-14}$$