Solve 6x^4y^5/18x^7y^8=

$$\frac { 6 x ^ { 4 } y ^ { 5 } } { 18 x ^ { 7 } y ^ { 8 } } =$$

Evaluate

$$\frac{1}{3\left(xy\right)^{3}}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{6^{1}x^{4}y^{5}}{18^{1}x^{7}y^{8}}$$

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

$$\frac{6^{1}}{18^{1}}x^{4-7}y^{5-8}$$

Subtract 7 from 4.

$$\frac{6^{1}}{18^{1}}x^{-3}y^{5-8}$$

Subtract 8 from 5.

$$\frac{6^{1}}{18^{1}}\times \frac{1}{x^{3}}y^{-3}$$

Reduce the fraction \frac{6}{18} to lowest terms by extracting and canceling out 6.

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

Differentiate w.r.t. x

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

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{6y^{5}}{18y^{8}}x^{4-7})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{3y^{3}}x^{-3})$$

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}.

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

Do the arithmetic.

$$\left(-\frac{1}{y^{3}}\right)x^{-4}$$