Solve 18x^7y^8/30x^6y^6

$$\frac { 18 x ^ { 7 } y ^ { 8 } } { 30 x ^ { 6 } y ^ { 6 } }$$

Evaluate

$$\frac{3xy^{2}}{5}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{18^{1}x^{7}y^{8}}{30^{1}x^{6}y^{6}}$$

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

$$\frac{18^{1}}{30^{1}}x^{7-6}y^{8-6}$$

Subtract 6 from 7.

$$\frac{18^{1}}{30^{1}}x^{1}y^{8-6}$$

Subtract 6 from 8.

$$\frac{18^{1}}{30^{1}}xy^{2}$$

Reduce the fraction \frac{18}{30} to lowest terms by extracting and canceling out 6.

$$\frac{3}{5}xy^{2}$$

Differentiate w.r.t. x

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

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{18y^{8}}{30y^{6}}x^{7-6})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3y^{2}}{5}x^{1})$$

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}.

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

Do the arithmetic.

$$\frac{3y^{2}}{5}x^{0}$$

For any term t except 0, t^{0}=1.

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

For any term t, t\times 1=t and 1t=t.

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