Solve 30y^7/6y^4

$$\frac { 30 y ^ { 7 } } { 6 y ^ { 4 } }$$

Evaluate

$$5y^{3}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(30y^{7}\right)^{1}\times \frac{1}{6y^{4}}$$

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

$$30^{1}\left(y^{7}\right)^{1}\times \frac{1}{6}\times \frac{1}{y^{4}}$$

Use the Commutative Property of Multiplication.

$$30^{1}\times \frac{1}{6}\left(y^{7}\right)^{1}\times \frac{1}{y^{4}}$$

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

$$30^{1}\times \frac{1}{6}y^{7}y^{4\left(-1\right)}$$

Multiply 4 times -1.

$$30^{1}\times \frac{1}{6}y^{7}y^{-4}$$

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

$$30^{1}\times \frac{1}{6}y^{7-4}$$

Add the exponents 7 and -4.

$$30^{1}\times \frac{1}{6}y^{3}$$

Raise 30 to the power 1.

$$30\times \frac{1}{6}y^{3}$$

Multiply 30 times \frac{1}{6}.

$$5y^{3}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{30^{1}y^{7}}{6^{1}y^{4}}$$

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

$$\frac{30^{1}y^{7-4}}{6^{1}}$$

Subtract 4 from 7.

$$\frac{30^{1}y^{3}}{6^{1}}$$

Divide 30 by 6.

$$5y^{3}$$

Differentiate w.r.t. y

$$15y^{2}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{30}{6}y^{7-4})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}y}(5y^{3})$$

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}.

$$3\times 5y^{3-1}$$

Do the arithmetic.

$$15y^{2}$$