Solve 16p^4/36p^9

$$\frac { 16 p ^ { 4 } } { 36 p ^ { 9 } }$$

Evaluate

$$\frac{4}{9p^{5}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(16p^{4}\right)^{1}\times \frac{1}{36p^{9}}$$

To raise the product of two or more numbers to a power, raise each number to the power and take their product.

$$16^{1}\left(p^{4}\right)^{1}\times \frac{1}{36}\times \frac{1}{p^{9}}$$

Use the Commutative Property of Multiplication.

$$16^{1}\times \frac{1}{36}\left(p^{4}\right)^{1}\times \frac{1}{p^{9}}$$

To raise a power to another power, multiply the exponents.

$$16^{1}\times \frac{1}{36}p^{4}p^{9\left(-1\right)}$$

Multiply 9 times -1.

$$16^{1}\times \frac{1}{36}p^{4}p^{-9}$$

To multiply powers of the same base, add their exponents.

$$16^{1}\times \frac{1}{36}p^{4-9}$$

Add the exponents 4 and -9.

$$16^{1}\times \frac{1}{36}p^{-5}$$

Raise 16 to the power 1.

$$16\times \frac{1}{36}p^{-5}$$

Multiply 16 times \frac{1}{36}.

$$\frac{4}{9}p^{-5}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{16^{1}p^{4}}{36^{1}p^{9}}$$

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

$$\frac{16^{1}p^{4-9}}{36^{1}}$$

Subtract 9 from 4.

$$\frac{16^{1}p^{-5}}{36^{1}}$$

Reduce the fraction \frac{16}{36} to lowest terms by extracting and canceling out 4.

$$\frac{4}{9}p^{-5}$$

Differentiate w.r.t. p

$$-\frac{20}{9p^{6}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}p}(\frac{16}{36}p^{4-9})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}p}(\frac{4}{9}p^{-5})$$

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}.

$$-5\times \frac{4}{9}p^{-5-1}$$

Do the arithmetic.

$$-\frac{20}{9}p^{-6}$$