Solve 18y^7/81y^3

$$\frac { 18 y ^ { 7 } } { 81 y ^ { 3 } }$$

Evaluate

$$\frac{2y^{4}}{9}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(18y^{7}\right)^{1}\times \frac{1}{81y^{3}}$$

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

$$18^{1}\left(y^{7}\right)^{1}\times \frac{1}{81}\times \frac{1}{y^{3}}$$

Use the Commutative Property of Multiplication.

$$18^{1}\times \frac{1}{81}\left(y^{7}\right)^{1}\times \frac{1}{y^{3}}$$

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

$$18^{1}\times \frac{1}{81}y^{7}y^{3\left(-1\right)}$$

Multiply 3 times -1.

$$18^{1}\times \frac{1}{81}y^{7}y^{-3}$$

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

$$18^{1}\times \frac{1}{81}y^{7-3}$$

Add the exponents 7 and -3.

$$18^{1}\times \frac{1}{81}y^{4}$$

Raise 18 to the power 1.

$$18\times \frac{1}{81}y^{4}$$

Multiply 18 times \frac{1}{81}.

$$\frac{2}{9}y^{4}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{18^{1}y^{7}}{81^{1}y^{3}}$$

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

$$\frac{18^{1}y^{7-3}}{81^{1}}$$

Subtract 3 from 7.

$$\frac{18^{1}y^{4}}{81^{1}}$$

Reduce the fraction \frac{18}{81} to lowest terms by extracting and canceling out 9.

$$\frac{2}{9}y^{4}$$

Differentiate w.r.t. y

$$\frac{8y^{3}}{9}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{18}{81}y^{7-3})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{2}{9}y^{4})$$

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}.

$$4\times \frac{2}{9}y^{4-1}$$

Do the arithmetic.

$$\frac{8}{9}y^{3}$$