Solve 4k^5/8k^2

$$\frac { 4 k ^ { 5 } } { 8 k ^ { 2 } }$$

Evaluate

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

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(4k^{5}\right)^{1}\times \frac{1}{8k^{2}}$$

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

$$4^{1}\left(k^{5}\right)^{1}\times \frac{1}{8}\times \frac{1}{k^{2}}$$

Use the Commutative Property of Multiplication.

$$4^{1}\times \frac{1}{8}\left(k^{5}\right)^{1}\times \frac{1}{k^{2}}$$

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

$$4^{1}\times \frac{1}{8}k^{5}k^{2\left(-1\right)}$$

Multiply 2 times -1.

$$4^{1}\times \frac{1}{8}k^{5}k^{-2}$$

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

$$4^{1}\times \frac{1}{8}k^{5-2}$$

Add the exponents 5 and -2.

$$4^{1}\times \frac{1}{8}k^{3}$$

Raise 4 to the power 1.

$$4\times \frac{1}{8}k^{3}$$

Multiply 4 times \frac{1}{8}.

$$\frac{1}{2}k^{3}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{4^{1}k^{5}}{8^{1}k^{2}}$$

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

$$\frac{4^{1}k^{5-2}}{8^{1}}$$

Subtract 2 from 5.

$$\frac{4^{1}k^{3}}{8^{1}}$$

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

$$\frac{1}{2}k^{3}$$

Differentiate w.r.t. k

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

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{4}{8}k^{5-2})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}k}(\frac{1}{2}k^{3})$$

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}.

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

Do the arithmetic.

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