Solve 20z^-7/5z^14

$$\frac { 20 z ^ { - 7 } } { 5 z ^ { 14 } }$$

Evaluate

$$\frac{4}{z^{21}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(20z^{-7}\right)^{1}\times \frac{1}{5z^{14}}$$

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

$$20^{1}\left(z^{-7}\right)^{1}\times \frac{1}{5}\times \frac{1}{z^{14}}$$

Use the Commutative Property of Multiplication.

$$20^{1}\times \frac{1}{5}\left(z^{-7}\right)^{1}\times \frac{1}{z^{14}}$$

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

$$20^{1}\times \frac{1}{5}z^{-7}z^{14\left(-1\right)}$$

Multiply 14 times -1.

$$20^{1}\times \frac{1}{5}z^{-7}z^{-14}$$

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

$$20^{1}\times \frac{1}{5}z^{-7-14}$$

Add the exponents -7 and -14.

$$20^{1}\times \frac{1}{5}z^{-21}$$

Raise 20 to the power 1.

$$20\times \frac{1}{5}z^{-21}$$

Multiply 20 times \frac{1}{5}.

$$4z^{-21}$$

Differentiate w.r.t. z

$$-\frac{84}{z^{22}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{20}{5}z^{-7-14})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}z}(4z^{-21})$$

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}.

$$-21\times 4z^{-21-1}$$

Do the arithmetic.

$$-84z^{-22}$$