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

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

Evaluate

$$-\frac{5}{a^{8}}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply -5 and 2 to get -10.

$$-5a^{2}a^{-10}$$

To multiply powers of the same base, add their exponents. Add 2 and -10 to get -8.

$$-5a^{-8}$$

Differentiate w.r.t. a

$$\frac{40}{a^{9}}$$

Steps Using Definition of a Derivative

To raise a power to another power, multiply the exponents. Multiply -5 and 2 to get -10.

$$\frac{\mathrm{d}}{\mathrm{d}a}(-5a^{2}a^{-10})$$

To multiply powers of the same base, add their exponents. Add 2 and -10 to get -8.

$$\frac{\mathrm{d}}{\mathrm{d}a}(-5a^{-8})$$

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

$$-8\left(-5\right)a^{-8-1}$$

Multiply -8 times -5.

$$40a^{-8-1}$$

Subtract 1 from -8.

$$40a^{-9}$$