Solve 12y^9/3y^2

$$\frac { 12 y ^ { 9 } } { 3 y ^ { 2 } }$$

Evaluate

$$4y^{7}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(12y^{9}\right)^{1}\times \frac{1}{3y^{2}}$$

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

$$12^{1}\left(y^{9}\right)^{1}\times \frac{1}{3}\times \frac{1}{y^{2}}$$

Use the Commutative Property of Multiplication.

$$12^{1}\times \frac{1}{3}\left(y^{9}\right)^{1}\times \frac{1}{y^{2}}$$

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

$$12^{1}\times \frac{1}{3}y^{9}y^{2\left(-1\right)}$$

Multiply 2 times -1.

$$12^{1}\times \frac{1}{3}y^{9}y^{-2}$$

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

$$12^{1}\times \frac{1}{3}y^{9-2}$$

Add the exponents 9 and -2.

$$12^{1}\times \frac{1}{3}y^{7}$$

Raise 12 to the power 1.

$$12\times \frac{1}{3}y^{7}$$

Multiply 12 times \frac{1}{3}.

$$4y^{7}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{12^{1}y^{9}}{3^{1}y^{2}}$$

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

$$\frac{12^{1}y^{9-2}}{3^{1}}$$

Subtract 2 from 9.

$$\frac{12^{1}y^{7}}{3^{1}}$$

Divide 12 by 3.

$$4y^{7}$$

Differentiate w.r.t. y

$$28y^{6}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{12}{3}y^{9-2})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}y}(4y^{7})$$

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}.

$$7\times 4y^{7-1}$$

Do the arithmetic.

$$28y^{6}$$