Solve 9z^8/10z^7

$$\frac { 9 z ^ { 8 } } { 10 z ^ { 7 } }$$

Evaluate

$$\frac{9z}{10}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(9z^{8}\right)^{1}\times \frac{1}{10z^{7}}$$

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

$$9^{1}\left(z^{8}\right)^{1}\times \frac{1}{10}\times \frac{1}{z^{7}}$$

Use the Commutative Property of Multiplication.

$$9^{1}\times \frac{1}{10}\left(z^{8}\right)^{1}\times \frac{1}{z^{7}}$$

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

$$9^{1}\times \frac{1}{10}z^{8}z^{7\left(-1\right)}$$

Multiply 7 times -1.

$$9^{1}\times \frac{1}{10}z^{8}z^{-7}$$

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

$$9^{1}\times \frac{1}{10}z^{8-7}$$

Add the exponents 8 and -7.

$$9^{1}\times \frac{1}{10}z^{1}$$

Raise 9 to the power 1.

$$9\times \frac{1}{10}z^{1}$$

Multiply 9 times \frac{1}{10}.

$$\frac{9}{10}z^{1}$$

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

$$\frac{9}{10}z$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{9^{1}z^{8}}{10^{1}z^{7}}$$

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

$$\frac{9^{1}z^{8-7}}{10^{1}}$$

Subtract 7 from 8.

$$\frac{9^{1}z^{1}}{10^{1}}$$

Divide 9 by 10.

$$\frac{9}{10}z^{1}$$

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

$$\frac{9}{10}z$$

Differentiate w.r.t. z

$$\frac{9}{10} = 0.9$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{9}{10}z^{8-7})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{9}{10}z^{1})$$

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}.

$$\frac{9}{10}z^{1-1}$$

Do the arithmetic.

$$\frac{9}{10}z^{0}$$

For any term t except 0, t^{0}=1.

$$\frac{9}{10}\times 1$$

For any term t, t\times 1=t and 1t=t.

$$\frac{9}{10}$$