Solve (a^4)^2*(a^5)^-2=

$$( a ^ { 4 } ) ^ { 2 } \cdot ( a ^ { 5 } ) ^ { - 2 } =$$

Evaluate

$$\frac{1}{a^{2}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(a^{4}\right)^{2}\left(a^{5}\right)^{-2}$$

To raise a power to another power, multiply the exponents.

$$a^{4\times 2}a^{5\left(-2\right)}$$

Multiply 4 times 2.

$$a^{8}a^{5\left(-2\right)}$$

Multiply 5 times -2.

$$a^{8}a^{-10}$$

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

$$a^{8-10}$$

Add the exponents 8 and -10.

$$a^{-2}$$

Differentiate w.r.t. a

$$-\frac{2}{a^{3}}$$

Steps Using Definition of a Derivative

To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{8}\left(a^{5}\right)^{-2})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{8}a^{-10})$$

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

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

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

$$-2a^{-2-1}$$

Subtract 1 from -2.

$$-2a^{-3}$$