Solve (a^7)^2*(a^7)^-3

$$( a ^ { 7 } ) ^ { 2 } \times ( a ^ { 7 } ) ^ { - 3 }$$

Evaluate

$$\frac{1}{a^{7}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(a^{7}\right)^{2}\left(a^{7}\right)^{-3}$$

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

$$a^{7\times 2}a^{7\left(-3\right)}$$

Multiply 7 times 2.

$$a^{14}a^{7\left(-3\right)}$$

Multiply 7 times -3.

$$a^{14}a^{-21}$$

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

$$a^{14-21}$$

Add the exponents 14 and -21.

$$a^{-7}$$

Differentiate w.r.t. a

$$-\frac{7}{a^{8}}$$

Steps Using Definition of a Derivative

To raise a power to another power, multiply the exponents. Multiply 7 and 2 to get 14.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{14}\left(a^{7}\right)^{-3})$$

To raise a power to another power, multiply the exponents. Multiply 7 and -3 to get -21.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{14}a^{-21})$$

To multiply powers of the same base, add their exponents. Add 14 and -21 to get -7.

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

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

$$-7a^{-7-1}$$

Subtract 1 from -7.

$$-7a^{-8}$$