Solve 4(z)^2/-20z^-6

$$\frac { 4 ( z ) ^ { 2 } } { - 20 z ^ { - 6 } }$$

Evaluate

$$-\frac{z^{8}}{5}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(4z^{2}\right)^{1}\times \frac{1}{-20z^{-6}}$$

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

$$4^{1}\left(z^{2}\right)^{1}\times \frac{1}{-20}\times \frac{1}{z^{-6}}$$

Use the Commutative Property of Multiplication.

$$4^{1}\times \frac{1}{-20}\left(z^{2}\right)^{1}\times \frac{1}{z^{-6}}$$

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

$$4^{1}\times \frac{1}{-20}z^{2}z^{-6\left(-1\right)}$$

Multiply -6 times -1.

$$4^{1}\times \frac{1}{-20}z^{2}z^{6}$$

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

$$4^{1}\times \frac{1}{-20}z^{2+6}$$

Add the exponents 2 and 6.

$$4^{1}\times \frac{1}{-20}z^{8}$$

Raise 4 to the power 1.

$$4\times \frac{1}{-20}z^{8}$$

Raise -20 to the power -1.

$$4\left(-\frac{1}{20}\right)z^{8}$$

Multiply 4 times -\frac{1}{20}.

$$-\frac{1}{5}z^{8}$$

Differentiate w.r.t. z

$$-\frac{8z^{7}}{5}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{4}{-20}z^{2-\left(-6\right)})$$

Do the arithmetic.

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

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}.

$$8\left(-\frac{1}{5}\right)z^{8-1}$$

Do the arithmetic.

$$-\frac{8}{5}z^{7}$$