Solve 15p^12/5p^8

$$\frac { 15 p ^ { 12 } } { 5 p ^ { 8 } }$$

Evaluate

$$3p^{4}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(15p^{12}\right)^{1}\times \frac{1}{5p^{8}}$$

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

$$15^{1}\left(p^{12}\right)^{1}\times \frac{1}{5}\times \frac{1}{p^{8}}$$

Use the Commutative Property of Multiplication.

$$15^{1}\times \frac{1}{5}\left(p^{12}\right)^{1}\times \frac{1}{p^{8}}$$

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

$$15^{1}\times \frac{1}{5}p^{12}p^{8\left(-1\right)}$$

Multiply 8 times -1.

$$15^{1}\times \frac{1}{5}p^{12}p^{-8}$$

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

$$15^{1}\times \frac{1}{5}p^{12-8}$$

Add the exponents 12 and -8.

$$15^{1}\times \frac{1}{5}p^{4}$$

Raise 15 to the power 1.

$$15\times \frac{1}{5}p^{4}$$

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

$$3p^{4}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{15^{1}p^{12}}{5^{1}p^{8}}$$

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

$$\frac{15^{1}p^{12-8}}{5^{1}}$$

Subtract 8 from 12.

$$\frac{15^{1}p^{4}}{5^{1}}$$

Divide 15 by 5.

$$3p^{4}$$

Differentiate w.r.t. p

$$12p^{3}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}p}(\frac{15}{5}p^{12-8})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}p}(3p^{4})$$

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}.

$$4\times 3p^{4-1}$$

Do the arithmetic.

$$12p^{3}$$