Solve 8w^2/36w^3

$$\frac { 8 w ^ { 2 } } { 36 w ^ { 3 } }$$

Evaluate

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

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(8w^{2}\right)^{1}\times \frac{1}{36w^{3}}$$

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

$$8^{1}\left(w^{2}\right)^{1}\times \frac{1}{36}\times \frac{1}{w^{3}}$$

Use the Commutative Property of Multiplication.

$$8^{1}\times \frac{1}{36}\left(w^{2}\right)^{1}\times \frac{1}{w^{3}}$$

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

$$8^{1}\times \frac{1}{36}w^{2}w^{3\left(-1\right)}$$

Multiply 3 times -1.

$$8^{1}\times \frac{1}{36}w^{2}w^{-3}$$

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

$$8^{1}\times \frac{1}{36}w^{2-3}$$

Add the exponents 2 and -3.

$$8^{1}\times \frac{1}{36}\times \frac{1}{w}$$

Raise 8 to the power 1.

$$8\times \frac{1}{36}\times \frac{1}{w}$$

Multiply 8 times \frac{1}{36}.

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

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{8^{1}w^{2}}{36^{1}w^{3}}$$

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

$$\frac{8^{1}w^{2-3}}{36^{1}}$$

Subtract 3 from 2.

$$\frac{8^{1}\times \frac{1}{w}}{36^{1}}$$

Reduce the fraction \frac{8}{36} to lowest terms by extracting and canceling out 4.

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

Differentiate w.r.t. w

$$-\frac{2}{9w^{2}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}w}(\frac{8}{36}w^{2-3})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}w}(\frac{2}{9}\times \frac{1}{w})$$

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}.

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

Do the arithmetic.

$$-\frac{2}{9}w^{-2}$$