Solve 16x^3y^4/40x^6y^8

$$\frac { 16 x ^ { 3 } y ^ { 4 } } { 40 x ^ { 6 } y ^ { 8 } }$$

Evaluate

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

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{16^{1}x^{3}y^{4}}{40^{1}x^{6}y^{8}}$$

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

$$\frac{16^{1}}{40^{1}}x^{3-6}y^{4-8}$$

Subtract 6 from 3.

$$\frac{16^{1}}{40^{1}}x^{-3}y^{4-8}$$

Subtract 8 from 4.

$$\frac{16^{1}}{40^{1}}\times \frac{1}{x^{3}}y^{-4}$$

Reduce the fraction \frac{16}{40} to lowest terms by extracting and canceling out 8.

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

Differentiate w.r.t. x

$$-\frac{6}{5\left(xy\right)^{4}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{16y^{4}}{40y^{8}}x^{3-6})$$

Do the arithmetic.

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

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}.

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

Do the arithmetic.

$$\left(-\frac{6}{5y^{4}}\right)x^{-4}$$