Solve 4x^1/2/8x^1/3

$$\frac { 4 x ^ { 1 / 2 } } { 8 x ^ { 1 / 3 } }$$

Evaluate

$$\frac{\sqrt[6]{x}}{2}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{4^{1}\sqrt{x}}{8^{1}\sqrt[3]{x}}$$

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

$$\frac{4^{1}x^{\frac{1}{2}-\frac{1}{3}}}{8^{1}}$$

Subtract \frac{1}{3} from \frac{1}{2} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$\frac{4^{1}\sqrt[6]{x}}{8^{1}}$$

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

$$\frac{1}{2}\sqrt[6]{x}$$

Differentiate w.r.t. x

$$\frac{1}{12x^{\frac{5}{6}}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4}{8}x^{\frac{1}{2}-\frac{1}{3}})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{2}\sqrt[6]{x})$$

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}.

$$\frac{1}{6}\times \left(\frac{1}{2}\right)x^{\frac{1}{6}-1}$$

Do the arithmetic.

$$\frac{1}{12}x^{-\frac{5}{6}}$$