Solve 16v^4/12v^7

$$\frac { 16 v ^ { 4 } } { 12 v ^ { 7 } }$$

Evaluate

$$\frac{4}{3v^{3}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(16v^{4}\right)^{1}\times \frac{1}{12v^{7}}$$

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

$$16^{1}\left(v^{4}\right)^{1}\times \frac{1}{12}\times \frac{1}{v^{7}}$$

Use the Commutative Property of Multiplication.

$$16^{1}\times \frac{1}{12}\left(v^{4}\right)^{1}\times \frac{1}{v^{7}}$$

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

$$16^{1}\times \frac{1}{12}v^{4}v^{7\left(-1\right)}$$

Multiply 7 times -1.

$$16^{1}\times \frac{1}{12}v^{4}v^{-7}$$

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

$$16^{1}\times \frac{1}{12}v^{4-7}$$

Add the exponents 4 and -7.

$$16^{1}\times \frac{1}{12}v^{-3}$$

Raise 16 to the power 1.

$$16\times \frac{1}{12}v^{-3}$$

Multiply 16 times \frac{1}{12}.

$$\frac{4}{3}v^{-3}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{16^{1}v^{4}}{12^{1}v^{7}}$$

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

$$\frac{16^{1}v^{4-7}}{12^{1}}$$

Subtract 7 from 4.

$$\frac{16^{1}v^{-3}}{12^{1}}$$

Reduce the fraction \frac{16}{12} to lowest terms by extracting and canceling out 4.

$$\frac{4}{3}v^{-3}$$

Differentiate w.r.t. v

$$-\frac{4}{v^{4}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}v}(\frac{16}{12}v^{4-7})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}v}(\frac{4}{3}v^{-3})$$

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}.

$$-3\times \frac{4}{3}v^{-3-1}$$

Do the arithmetic.

$$-4v^{-4}$$