Solve (-3)^4p^2q^5/(-3)pq

$$\frac { ( - 3 ) ^ { 4 } p ^ { 2 } q ^ { 5 } } { ( - 3 ) p q }$$

Evaluate

$$-27pq^{4}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{\left(-3\right)^{4}p^{2}q^{5}}{\left(-3\right)^{1}p^{1}q^{1}}$$

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

$$\left(-3\right)^{4-1}p^{2-1}q^{5-1}$$

Subtract 1 from 4.

$$\left(-3\right)^{3}p^{2-1}q^{5-1}$$

Subtract 1 from 2.

$$-27p^{1}q^{5-1}$$

Subtract 1 from 5.

$$-27p^{1}q^{4}$$

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

$$-27pq^{4}$$

Differentiate w.r.t. p

$$-27q^{4}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}p}(\frac{81q^{5}}{-3q}p^{2-1})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}p}(\left(-27q^{4}\right)p^{1})$$

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}.

$$\left(-27q^{4}\right)p^{1-1}$$

Do the arithmetic.

$$\left(-27q^{4}\right)p^{0}$$

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

$$\left(-27q^{4}\right)\times 1$$

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

$$-27q^{4}$$