Solve 42n^9/6n^4

$$\frac { 42 n ^ { 9 } } { 6 n ^ { 4 } }$$

Evaluate

$$7n^{5}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(42n^{9}\right)^{1}\times \frac{1}{6n^{4}}$$

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

$$42^{1}\left(n^{9}\right)^{1}\times \frac{1}{6}\times \frac{1}{n^{4}}$$

Use the Commutative Property of Multiplication.

$$42^{1}\times \frac{1}{6}\left(n^{9}\right)^{1}\times \frac{1}{n^{4}}$$

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

$$42^{1}\times \frac{1}{6}n^{9}n^{4\left(-1\right)}$$

Multiply 4 times -1.

$$42^{1}\times \frac{1}{6}n^{9}n^{-4}$$

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

$$42^{1}\times \frac{1}{6}n^{9-4}$$

Add the exponents 9 and -4.

$$42^{1}\times \frac{1}{6}n^{5}$$

Raise 42 to the power 1.

$$42\times \frac{1}{6}n^{5}$$

Multiply 42 times \frac{1}{6}.

$$7n^{5}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{42^{1}n^{9}}{6^{1}n^{4}}$$

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

$$\frac{42^{1}n^{9-4}}{6^{1}}$$

Subtract 4 from 9.

$$\frac{42^{1}n^{5}}{6^{1}}$$

Divide 42 by 6.

$$7n^{5}$$

Differentiate w.r.t. n

$$35n^{4}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{42}{6}n^{9-4})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}n}(7n^{5})$$

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}.

$$5\times 7n^{5-1}$$

Do the arithmetic.

$$35n^{4}$$