Solve 16k^-2/4k^6

$$\frac { 16 k ^ { - 2 } } { 4 k ^ { 6 } }$$

Evaluate

$$\frac{4}{k^{8}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(16k^{-2}\right)^{1}\times \frac{1}{4k^{6}}$$

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

$$16^{1}\left(k^{-2}\right)^{1}\times \frac{1}{4}\times \frac{1}{k^{6}}$$

Use the Commutative Property of Multiplication.

$$16^{1}\times \frac{1}{4}\left(k^{-2}\right)^{1}\times \frac{1}{k^{6}}$$

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

$$16^{1}\times \frac{1}{4}k^{-2}k^{6\left(-1\right)}$$

Multiply 6 times -1.

$$16^{1}\times \frac{1}{4}k^{-2}k^{-6}$$

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

$$16^{1}\times \frac{1}{4}k^{-2-6}$$

Add the exponents -2 and -6.

$$16^{1}\times \frac{1}{4}k^{-8}$$

Raise 16 to the power 1.

$$16\times \frac{1}{4}k^{-8}$$

Multiply 16 times \frac{1}{4}.

$$4k^{-8}$$

Differentiate w.r.t. k

$$-\frac{32}{k^{9}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{16}{4}k^{-2-6})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}k}(4k^{-8})$$

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}.

$$-8\times 4k^{-8-1}$$

Do the arithmetic.

$$-32k^{-9}$$