Solve 30z^5/42z^3

$$\frac { 30 z ^ { 5 } } { 42 z ^ { 3 } }$$

Evaluate

$$\frac{5z^{2}}{7}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(30z^{5}\right)^{1}\times \frac{1}{42z^{3}}$$

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

$$30^{1}\left(z^{5}\right)^{1}\times \frac{1}{42}\times \frac{1}{z^{3}}$$

Use the Commutative Property of Multiplication.

$$30^{1}\times \frac{1}{42}\left(z^{5}\right)^{1}\times \frac{1}{z^{3}}$$

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

$$30^{1}\times \frac{1}{42}z^{5}z^{3\left(-1\right)}$$

Multiply 3 times -1.

$$30^{1}\times \frac{1}{42}z^{5}z^{-3}$$

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

$$30^{1}\times \frac{1}{42}z^{5-3}$$

Add the exponents 5 and -3.

$$30^{1}\times \frac{1}{42}z^{2}$$

Raise 30 to the power 1.

$$30\times \frac{1}{42}z^{2}$$

Multiply 30 times \frac{1}{42}.

$$\frac{5}{7}z^{2}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{30^{1}z^{5}}{42^{1}z^{3}}$$

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

$$\frac{30^{1}z^{5-3}}{42^{1}}$$

Subtract 3 from 5.

$$\frac{30^{1}z^{2}}{42^{1}}$$

Reduce the fraction \frac{30}{42} to lowest terms by extracting and canceling out 6.

$$\frac{5}{7}z^{2}$$

Differentiate w.r.t. z

$$\frac{10z}{7}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{30}{42}z^{5-3})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{5}{7}z^{2})$$

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}.

$$2\times \frac{5}{7}z^{2-1}$$

Do the arithmetic.

$$\frac{10}{7}z^{1}$$

For any term t, t^{1}=t.

$$\frac{10}{7}z$$