Solve 6n^2/3n^5

$$\frac { 6 n ^ { 2 } } { 3 n ^ { 5 } }$$

Evaluate

$$\frac{2}{n^{3}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(6n^{2}\right)^{1}\times \frac{1}{3n^{5}}$$

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

$$6^{1}\left(n^{2}\right)^{1}\times \frac{1}{3}\times \frac{1}{n^{5}}$$

Use the Commutative Property of Multiplication.

$$6^{1}\times \frac{1}{3}\left(n^{2}\right)^{1}\times \frac{1}{n^{5}}$$

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

$$6^{1}\times \frac{1}{3}n^{2}n^{5\left(-1\right)}$$

Multiply 5 times -1.

$$6^{1}\times \frac{1}{3}n^{2}n^{-5}$$

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

$$6^{1}\times \frac{1}{3}n^{2-5}$$

Add the exponents 2 and -5.

$$6^{1}\times \frac{1}{3}n^{-3}$$

Raise 6 to the power 1.

$$6\times \frac{1}{3}n^{-3}$$

Multiply 6 times \frac{1}{3}.

$$2n^{-3}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{6^{1}n^{2}}{3^{1}n^{5}}$$

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

$$\frac{6^{1}n^{2-5}}{3^{1}}$$

Subtract 5 from 2.

$$\frac{6^{1}n^{-3}}{3^{1}}$$

Divide 6 by 3.

$$2n^{-3}$$

Differentiate w.r.t. n

$$-\frac{6}{n^{4}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{6}{3}n^{2-5})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}n}(2n^{-3})$$

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}.

$$-3\times 2n^{-3-1}$$

Do the arithmetic.

$$-6n^{-4}$$