Solve frac{z^frac{3{2}}}{z^frac{5{2}}}

$$\frac { z ^ { \frac { 3 } { 2 } } } { z ^ { \frac { 5 } { 2 } } }$$

Evaluate

$$\frac{1}{z}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{z^{\frac{3}{2}}}{z^{\frac{5}{2}}}$$

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

$$z^{\frac{3}{2}-\frac{5}{2}}$$

Subtract \frac{5}{2} from \frac{3}{2} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$\frac{1}{z}$$

Differentiate w.r.t. z

$$-\frac{1}{z^{2}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{1}{1}z^{\frac{3}{2}-\frac{5}{2}})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{1}{z})$$

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}.

$$-z^{-1-1}$$

Do the arithmetic.

$$-z^{-2}$$