Solve -27x^7y^2/-9x^6y^5

$$\frac { - 27 x ^ { 7 } y ^ { 2 } } { - 9 x ^ { 6 } y ^ { 5 } }$$

Evaluate

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

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{\left(-27\right)^{1}x^{7}y^{2}}{\left(-9\right)^{1}x^{6}y^{5}}$$

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

$$\frac{\left(-27\right)^{1}}{\left(-9\right)^{1}}x^{7-6}y^{2-5}$$

Subtract 6 from 7.

$$\frac{\left(-27\right)^{1}}{\left(-9\right)^{1}}x^{1}y^{2-5}$$

Subtract 5 from 2.

$$\frac{\left(-27\right)^{1}}{\left(-9\right)^{1}}xy^{-3}$$

Divide -27 by -9.

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

Differentiate w.r.t. x

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

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(-\frac{27y^{2}}{-9y^{5}}\right)x^{7-6})$$

Do the arithmetic.

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

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}.

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

Do the arithmetic.

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

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

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

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

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