Solve =8a^-12/a^5

$$= \frac { 8 a ^ { - 12 } } { a ^ { 5 } }$$

Evaluate

$$\frac{8}{a^{17}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(8a^{-12}\right)^{1}\times \frac{1}{a^{5}}$$

To raise the product of two or more numbers to a power, raise each number to the power and take their product.

$$8^{1}\left(a^{-12}\right)^{1}\times \frac{1}{1}\times \frac{1}{a^{5}}$$

Use the Commutative Property of Multiplication.

$$8^{1}\times \frac{1}{1}\left(a^{-12}\right)^{1}\times \frac{1}{a^{5}}$$

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

$$8^{1}\times \frac{1}{1}a^{-12}a^{5\left(-1\right)}$$

Multiply 5 times -1.

$$8^{1}\times \frac{1}{1}a^{-12}a^{-5}$$

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

$$8^{1}\times \frac{1}{1}a^{-12-5}$$

Add the exponents -12 and -5.

$$8^{1}\times \frac{1}{1}a^{-17}$$

Raise 8 to the power 1.

$$8\times \frac{1}{1}a^{-17}$$

Differentiate w.r.t. a

$$-\frac{136}{a^{18}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{8}{1}a^{-12-5})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}a}(8a^{-17})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$-17\times 8a^{-17-1}$$

Do the arithmetic.

$$-136a^{-18}$$