Solve 24x^7y^2/16x^3y^4

$$\frac { 24 x ^ { 7 } y ^ { 2 } } { 16 x ^ { 3 } y ^ { 4 } }$$

Evaluate

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

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{24^{1}x^{7}y^{2}}{16^{1}x^{3}y^{4}}$$

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

$$\frac{24^{1}}{16^{1}}x^{7-3}y^{2-4}$$

Subtract 3 from 7.

$$\frac{24^{1}}{16^{1}}x^{4}y^{2-4}$$

Subtract 4 from 2.

$$\frac{24^{1}}{16^{1}}x^{4}y^{-2}$$

Reduce the fraction \frac{24}{16} to lowest terms by extracting and canceling out 8.

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

Differentiate w.r.t. x

$$\frac{6x^{3}}{y^{2}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{24y^{2}}{16y^{4}}x^{7-3})$$

Do the arithmetic.

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

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}.

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

Do the arithmetic.

$$\frac{6}{y^{2}}x^{3}$$