Solve 28y^3/7y^7

$$\frac { 28 y ^ { 3 } } { 7 y ^ { 7 } }$$

Evaluate

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

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(28y^{3}\right)^{1}\times \frac{1}{7y^{7}}$$

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

$$28^{1}\left(y^{3}\right)^{1}\times \frac{1}{7}\times \frac{1}{y^{7}}$$

Use the Commutative Property of Multiplication.

$$28^{1}\times \frac{1}{7}\left(y^{3}\right)^{1}\times \frac{1}{y^{7}}$$

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

$$28^{1}\times \frac{1}{7}y^{3}y^{7\left(-1\right)}$$

Multiply 7 times -1.

$$28^{1}\times \frac{1}{7}y^{3}y^{-7}$$

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

$$28^{1}\times \frac{1}{7}y^{3-7}$$

Add the exponents 3 and -7.

$$28^{1}\times \frac{1}{7}y^{-4}$$

Raise 28 to the power 1.

$$28\times \frac{1}{7}y^{-4}$$

Multiply 28 times \frac{1}{7}.

$$4y^{-4}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{28^{1}y^{3}}{7^{1}y^{7}}$$

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

$$\frac{28^{1}y^{3-7}}{7^{1}}$$

Subtract 7 from 3.

$$\frac{28^{1}y^{-4}}{7^{1}}$$

Divide 28 by 7.

$$4y^{-4}$$

Differentiate w.r.t. y

$$-\frac{16}{y^{5}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{28}{7}y^{3-7})$$

Do the arithmetic.

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

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}.

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

Do the arithmetic.

$$-16y^{-5}$$