Solve 8c^5/12c^2

$$\frac { 8 c ^ { 5 } } { 12 c ^ { 2 } }$$

Evaluate

$$\frac{2c^{3}}{3}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(8c^{5}\right)^{1}\times \frac{1}{12c^{2}}$$

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

$$8^{1}\left(c^{5}\right)^{1}\times \frac{1}{12}\times \frac{1}{c^{2}}$$

Use the Commutative Property of Multiplication.

$$8^{1}\times \frac{1}{12}\left(c^{5}\right)^{1}\times \frac{1}{c^{2}}$$

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

$$8^{1}\times \frac{1}{12}c^{5}c^{2\left(-1\right)}$$

Multiply 2 times -1.

$$8^{1}\times \frac{1}{12}c^{5}c^{-2}$$

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

$$8^{1}\times \frac{1}{12}c^{5-2}$$

Add the exponents 5 and -2.

$$8^{1}\times \frac{1}{12}c^{3}$$

Raise 8 to the power 1.

$$8\times \frac{1}{12}c^{3}$$

Multiply 8 times \frac{1}{12}.

$$\frac{2}{3}c^{3}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{8^{1}c^{5}}{12^{1}c^{2}}$$

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

$$\frac{8^{1}c^{5-2}}{12^{1}}$$

Subtract 2 from 5.

$$\frac{8^{1}c^{3}}{12^{1}}$$

Reduce the fraction \frac{8}{12} to lowest terms by extracting and canceling out 4.

$$\frac{2}{3}c^{3}$$

Differentiate w.r.t. c

$$2c^{2}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}c}(\frac{8}{12}c^{5-2})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}c}(\frac{2}{3}c^{3})$$

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}.

$$3\times \frac{2}{3}c^{3-1}$$

Do the arithmetic.

$$2c^{2}$$