Solve 3n^-1/5n^-4

$$\frac { 3 n ^ { - 1 } } { 5 n ^ { - 4 } }$$

Evaluate

$$\frac{3n^{3}}{5}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(3\times \frac{1}{n}\right)^{1}\times \frac{1}{5n^{-4}}$$

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

$$3^{1}\times \left(\frac{1}{n}\right)^{1}\times \frac{1}{5}\times \frac{1}{n^{-4}}$$

Use the Commutative Property of Multiplication.

$$3^{1}\times \frac{1}{5}\times \left(\frac{1}{n}\right)^{1}\times \frac{1}{n^{-4}}$$

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

$$3^{1}\times \frac{1}{5}\times \frac{1}{n}n^{-4\left(-1\right)}$$

Multiply -4 times -1.

$$3^{1}\times \frac{1}{5}\times \frac{1}{n}n^{4}$$

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

$$3^{1}\times \frac{1}{5}n^{-1+4}$$

Add the exponents -1 and 4.

$$3^{1}\times \frac{1}{5}n^{3}$$

Raise 3 to the power 1.

$$3\times \frac{1}{5}n^{3}$$

Multiply 3 times \frac{1}{5}.

$$\frac{3}{5}n^{3}$$

Differentiate w.r.t. n

$$\frac{9n^{2}}{5}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{3}{5}n^{-1-\left(-4\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{3}{5}n^{3})$$

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}.

$$3\times \frac{3}{5}n^{3-1}$$

Do the arithmetic.

$$\frac{9}{5}n^{2}$$