Solve x^2/4x^frac{5{3}}

$$\frac { x ^ { 2 } } { 4 x ^ { \frac { 5 } { 3 } } }$$

Evaluate

$$\frac{\sqrt[3]{x}}{4}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(x^{1}\right)^{2}\times \frac{1}{4x^{\frac{5}{3}}}$$

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

$$1^{2}\left(x^{1}\right)^{2}\times \frac{1}{4}\times \frac{1}{x^{\frac{5}{3}}}$$

Use the Commutative Property of Multiplication.

$$1^{2}\times \frac{1}{4}\left(x^{1}\right)^{2}\times \frac{1}{x^{\frac{5}{3}}}$$

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

$$1^{2}\times \frac{1}{4}x^{2}x^{\frac{5}{3}\left(-1\right)}$$

Multiply \frac{5}{3} times -1.

$$1^{2}\times \frac{1}{4}x^{2}x^{-\frac{5}{3}}$$

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

$$1^{2}\times \frac{1}{4}x^{2-\frac{5}{3}}$$

Add the exponents 2 and -\frac{5}{3}.

$$1^{2}\times \frac{1}{4}\sqrt[3]{x}$$

Raise 4 to the power -1.

$$\frac{1}{4}\sqrt[3]{x}$$

Differentiate w.r.t. x

$$\frac{1}{12x^{\frac{2}{3}}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{4}x^{2-\frac{5}{3}})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{4}\sqrt[3]{x})$$

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}.

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

Do the arithmetic.

$$\frac{1}{12}x^{-\frac{2}{3}}$$