Solve 52z^4/13z^6

$$\frac { 52 z ^ { 4 } } { 13 z ^ { 6 } }$$

Evaluate

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

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(52z^{4}\right)^{1}\times \frac{1}{13z^{6}}$$

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

$$52^{1}\left(z^{4}\right)^{1}\times \frac{1}{13}\times \frac{1}{z^{6}}$$

Use the Commutative Property of Multiplication.

$$52^{1}\times \frac{1}{13}\left(z^{4}\right)^{1}\times \frac{1}{z^{6}}$$

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

$$52^{1}\times \frac{1}{13}z^{4}z^{6\left(-1\right)}$$

Multiply 6 times -1.

$$52^{1}\times \frac{1}{13}z^{4}z^{-6}$$

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

$$52^{1}\times \frac{1}{13}z^{4-6}$$

Add the exponents 4 and -6.

$$52^{1}\times \frac{1}{13}z^{-2}$$

Raise 52 to the power 1.

$$52\times \frac{1}{13}z^{-2}$$

Multiply 52 times \frac{1}{13}.

$$4z^{-2}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{52^{1}z^{4}}{13^{1}z^{6}}$$

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

$$\frac{52^{1}z^{4-6}}{13^{1}}$$

Subtract 6 from 4.

$$\frac{52^{1}z^{-2}}{13^{1}}$$

Divide 52 by 13.

$$4z^{-2}$$

Differentiate w.r.t. z

$$-\frac{8}{z^{3}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{52}{13}z^{4-6})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}z}(4z^{-2})$$

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}.

$$-2\times 4z^{-2-1}$$

Do the arithmetic.

$$-8z^{-3}$$