Solve (15p^6)div(5p^-3)

$$( 15 p ^ { 6 } ) \div ( 5 p ^ { - 3 } )$$

Evaluate

$$3p^{9}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(15p^{6}\right)^{1}\times \frac{1}{5p^{-3}}$$

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

$$15^{1}\left(p^{6}\right)^{1}\times \frac{1}{5}\times \frac{1}{p^{-3}}$$

Use the Commutative Property of Multiplication.

$$15^{1}\times \frac{1}{5}\left(p^{6}\right)^{1}\times \frac{1}{p^{-3}}$$

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

$$15^{1}\times \frac{1}{5}p^{6}p^{-3\left(-1\right)}$$

Multiply -3 times -1.

$$15^{1}\times \frac{1}{5}p^{6}p^{3}$$

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

$$15^{1}\times \frac{1}{5}p^{6+3}$$

Add the exponents 6 and 3.

$$15^{1}\times \frac{1}{5}p^{9}$$

Raise 15 to the power 1.

$$15\times \frac{1}{5}p^{9}$$

Multiply 15 times \frac{1}{5}.

$$3p^{9}$$

Differentiate w.r.t. p

$$27p^{8}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}p}(\frac{15}{5}p^{6-\left(-3\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}p}(3p^{9})$$

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}.

$$9\times 3p^{9-1}$$

Do the arithmetic.

$$27p^{8}$$