Solve 8x^4y^3/12x^8y

$$\frac { 8 x ^ { 4 } y ^ { 3 } } { 12 x ^ { 8 } y }$$

Evaluate

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

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{8^{1}x^{4}y^{3}}{12^{1}x^{8}y^{1}}$$

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

$$\frac{8^{1}}{12^{1}}x^{4-8}y^{3-1}$$

Subtract 8 from 4.

$$\frac{8^{1}}{12^{1}}x^{-4}y^{3-1}$$

Subtract 1 from 3.

$$\frac{8^{1}}{12^{1}}\times \frac{1}{x^{4}}y^{2}$$

Reduce the fraction \frac{8}{12} to lowest terms by extracting and canceling out 4.

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

Differentiate w.r.t. x

$$-\frac{8y^{2}}{3x^{5}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{8y^{3}}{12y}x^{4-8})$$

Do the arithmetic.

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

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}.

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

Do the arithmetic.

$$\left(-\frac{8y^{2}}{3}\right)x^{-5}$$