Solve 9x^8y^3/6x^7y^7

$$\frac { 9 x ^ { 8 } y ^ { 3 } } { 6 x ^ { 7 } y ^ { 7 } }$$

Evaluate

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

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{9^{1}x^{8}y^{3}}{6^{1}x^{7}y^{7}}$$

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

$$\frac{9^{1}}{6^{1}}x^{8-7}y^{3-7}$$

Subtract 7 from 8.

$$\frac{9^{1}}{6^{1}}x^{1}y^{3-7}$$

Subtract 7 from 3.

$$\frac{9^{1}}{6^{1}}xy^{-4}$$

Reduce the fraction \frac{9}{6} to lowest terms by extracting and canceling out 3.

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

Differentiate w.r.t. x

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

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{9y^{3}}{6y^{7}}x^{8-7})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3}{2y^{4}}x^{1})$$

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}.

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

Do the arithmetic.

$$\frac{3}{2y^{4}}x^{0}$$

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

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

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

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