Solve 2r^9/9r^8

$$\frac { 2 r ^ { 9 } } { 9 r ^ { 8 } }$$

Evaluate

$$\frac{2r}{9}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(2r^{9}\right)^{1}\times \frac{1}{9r^{8}}$$

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

$$2^{1}\left(r^{9}\right)^{1}\times \frac{1}{9}\times \frac{1}{r^{8}}$$

Use the Commutative Property of Multiplication.

$$2^{1}\times \frac{1}{9}\left(r^{9}\right)^{1}\times \frac{1}{r^{8}}$$

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

$$2^{1}\times \frac{1}{9}r^{9}r^{8\left(-1\right)}$$

Multiply 8 times -1.

$$2^{1}\times \frac{1}{9}r^{9}r^{-8}$$

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

$$2^{1}\times \frac{1}{9}r^{9-8}$$

Add the exponents 9 and -8.

$$2^{1}\times \frac{1}{9}r^{1}$$

Raise 2 to the power 1.

$$2\times \frac{1}{9}r^{1}$$

Multiply 2 times \frac{1}{9}.

$$\frac{2}{9}r^{1}$$

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

$$\frac{2}{9}r$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{2^{1}r^{9}}{9^{1}r^{8}}$$

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

$$\frac{2^{1}r^{9-8}}{9^{1}}$$

Subtract 8 from 9.

$$\frac{2^{1}r^{1}}{9^{1}}$$

Divide 2 by 9.

$$\frac{2}{9}r^{1}$$

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

$$\frac{2}{9}r$$

Differentiate w.r.t. r

$$\frac{2}{9} = 0.2222222222222222$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}r}(\frac{2}{9}r^{9-8})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}r}(\frac{2}{9}r^{1})$$

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}.

$$\frac{2}{9}r^{1-1}$$

Do the arithmetic.

$$\frac{2}{9}r^{0}$$

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

$$\frac{2}{9}\times 1$$

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

$$\frac{2}{9}$$