Solve -9n^8/27n^10

$$\frac { - 9 n ^ { 8 } } { 27 n ^ { 10 } }$$

Evaluate

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

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(-9n^{8}\right)^{1}\times \frac{1}{27n^{10}}$$

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

$$\left(-9\right)^{1}\left(n^{8}\right)^{1}\times \frac{1}{27}\times \frac{1}{n^{10}}$$

Use the Commutative Property of Multiplication.

$$\left(-9\right)^{1}\times \frac{1}{27}\left(n^{8}\right)^{1}\times \frac{1}{n^{10}}$$

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

$$\left(-9\right)^{1}\times \frac{1}{27}n^{8}n^{10\left(-1\right)}$$

Multiply 10 times -1.

$$\left(-9\right)^{1}\times \frac{1}{27}n^{8}n^{-10}$$

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

$$\left(-9\right)^{1}\times \frac{1}{27}n^{8-10}$$

Add the exponents 8 and -10.

$$\left(-9\right)^{1}\times \frac{1}{27}n^{-2}$$

Raise -9 to the power 1.

$$-9\times \frac{1}{27}n^{-2}$$

Multiply -9 times \frac{1}{27}.

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

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{\left(-9\right)^{1}n^{8}}{27^{1}n^{10}}$$

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

$$\frac{\left(-9\right)^{1}n^{8-10}}{27^{1}}$$

Subtract 10 from 8.

$$\frac{\left(-9\right)^{1}n^{-2}}{27^{1}}$$

Reduce the fraction \frac{-9}{27} to lowest terms by extracting and canceling out 9.

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

Differentiate w.r.t. n

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

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}n}(\left(-\frac{9}{27}\right)n^{8-10})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}n}(-\frac{1}{3}n^{-2})$$

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}.

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

Do the arithmetic.

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