Solve (z^-4/4z^-7)

$$( \frac { z ^ { - 4 } } { 4 z ^ { - 7 } } )$$

Evaluate

$$\frac{z^{3}}{4}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(z^{1}\right)^{-4}\times \frac{1}{4z^{-7}}$$

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

$$1^{-4}\left(z^{1}\right)^{-4}\times \frac{1}{4}\times \frac{1}{z^{-7}}$$

Use the Commutative Property of Multiplication.

$$1^{-4}\times \frac{1}{4}\left(z^{1}\right)^{-4}\times \frac{1}{z^{-7}}$$

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

$$1^{-4}\times \frac{1}{4}z^{-4}z^{-7\left(-1\right)}$$

Multiply -7 times -1.

$$1^{-4}\times \frac{1}{4}z^{-4}z^{7}$$

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

$$1^{-4}\times \frac{1}{4}z^{-4+7}$$

Add the exponents -4 and 7.

$$1^{-4}\times \frac{1}{4}z^{3}$$

Raise 4 to the power -1.

$$\frac{1}{4}z^{3}$$

Differentiate w.r.t. z

$$\frac{3z^{2}}{4}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{1}{4}z^{-4-\left(-7\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{1}{4}z^{3})$$

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}.

$$3\times \frac{1}{4}z^{3-1}$$

Do the arithmetic.

$$\frac{3}{4}z^{2}$$