Solve 45w^7/5w^3

$$\frac { 45 w ^ { 7 } } { 5 w ^ { 3 } }$$

Evaluate

$$9w^{4}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(45w^{7}\right)^{1}\times \frac{1}{5w^{3}}$$

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

$$45^{1}\left(w^{7}\right)^{1}\times \frac{1}{5}\times \frac{1}{w^{3}}$$

Use the Commutative Property of Multiplication.

$$45^{1}\times \frac{1}{5}\left(w^{7}\right)^{1}\times \frac{1}{w^{3}}$$

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

$$45^{1}\times \frac{1}{5}w^{7}w^{3\left(-1\right)}$$

Multiply 3 times -1.

$$45^{1}\times \frac{1}{5}w^{7}w^{-3}$$

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

$$45^{1}\times \frac{1}{5}w^{7-3}$$

Add the exponents 7 and -3.

$$45^{1}\times \frac{1}{5}w^{4}$$

Raise 45 to the power 1.

$$45\times \frac{1}{5}w^{4}$$

Multiply 45 times \frac{1}{5}.

$$9w^{4}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{45^{1}w^{7}}{5^{1}w^{3}}$$

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

$$\frac{45^{1}w^{7-3}}{5^{1}}$$

Subtract 3 from 7.

$$\frac{45^{1}w^{4}}{5^{1}}$$

Divide 45 by 5.

$$9w^{4}$$

Differentiate w.r.t. w

$$36w^{3}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}w}(\frac{45}{5}w^{7-3})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}w}(9w^{4})$$

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}.

$$4\times 9w^{4-1}$$

Do the arithmetic.

$$36w^{3}$$