Solve frac{(a^{2})^{4}}{b^-1}div(a^4)^-2/b^-1

$$\frac { ( a ^ { 2 } ) ^ { 4 } } { b ^ { - 1 } } \div \frac { ( a ^ { 4 } ) ^ { - 2 } } { b ^ { - 1 } }$$

Evaluate

$$a^{16}$$

Solution Steps

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

$$\frac{\frac{a^{8}}{b^{-1}}}{\frac{\left(a^{4}\right)^{-2}}{b^{-1}}}$$

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

$$\frac{\frac{a^{8}}{b^{-1}}}{\frac{a^{-8}}{b^{-1}}}$$

Divide \frac{a^{8}}{b^{-1}} by \frac{a^{-8}}{b^{-1}} by multiplying \frac{a^{8}}{b^{-1}} by the reciprocal of \frac{a^{-8}}{b^{-1}}.

$$\frac{a^{8}b^{-1}}{b^{-1}a^{-8}}$$

Cancel out \frac{1}{b} in both numerator and denominator.

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

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$a^{16}$$

Differentiate w.r.t. a

$$16a^{15}$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\frac{a^{8}}{b^{-1}}}{\frac{\left(a^{4}\right)^{-2}}{b^{-1}}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{\frac{a^{8}}{b^{-1}}}{\frac{a^{-8}}{b^{-1}}})$$

Divide \frac{a^{8}}{b^{-1}} by \frac{a^{-8}}{b^{-1}} by multiplying \frac{a^{8}}{b^{-1}} by the reciprocal of \frac{a^{-8}}{b^{-1}}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{8}b^{-1}}{b^{-1}a^{-8}})$$

Cancel out \frac{1}{b} in both numerator and denominator.

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

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{\mathrm{d}}{\mathrm{d}a}(a^{16})$$

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

$$16a^{16-1}$$

Subtract 1 from 16.

$$16a^{15}$$