Solve 24x^2y^5/6x^frac{4{3}y^frac{3{4}}}

$$\frac { 24 x ^ { 2 } y ^ { 5 } } { 6 x ^ { \frac { 4 } { 3 } } y ^ { \frac { 3 } { 4 } } }$$

Evaluate

$$4x^{\frac{2}{3}}y^{\frac{17}{4}}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{24^{1}x^{2}y^{5}}{6^{1}x^{\frac{4}{3}}y^{\frac{3}{4}}}$$

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

$$\frac{24^{1}}{6^{1}}x^{2-\frac{4}{3}}y^{5-\frac{3}{4}}$$

Subtract \frac{4}{3} from 2.

$$\frac{24^{1}}{6^{1}}x^{\frac{2}{3}}y^{5-\frac{3}{4}}$$

Subtract \frac{3}{4} from 5.

$$\frac{24^{1}}{6^{1}}x^{\frac{2}{3}}y^{\frac{17}{4}}$$

Divide 24 by 6.

$$4x^{\frac{2}{3}}y^{\frac{17}{4}}$$

Differentiate w.r.t. x

$$\frac{8y^{\frac{17}{4}}}{3\sqrt[3]{x}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{24y^{5}}{6y^{\frac{3}{4}}}x^{2-\frac{4}{3}})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(4y^{\frac{17}{4}}x^{\frac{2}{3}})$$

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}.

$$\frac{2}{3}\times 4y^{\frac{17}{4}}x^{\frac{2}{3}-1}$$

Do the arithmetic.

$$\frac{8y^{\frac{17}{4}}}{3}x^{-\frac{1}{3}}$$