Solve 8q^4/14q^10

$$\frac { 8 q ^ { 4 } } { 14 q ^ { 10 } }$$

Evaluate

$$\frac{4}{7q^{6}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(8q^{4}\right)^{1}\times \frac{1}{14q^{10}}$$

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

$$8^{1}\left(q^{4}\right)^{1}\times \frac{1}{14}\times \frac{1}{q^{10}}$$

Use the Commutative Property of Multiplication.

$$8^{1}\times \frac{1}{14}\left(q^{4}\right)^{1}\times \frac{1}{q^{10}}$$

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

$$8^{1}\times \frac{1}{14}q^{4}q^{10\left(-1\right)}$$

Multiply 10 times -1.

$$8^{1}\times \frac{1}{14}q^{4}q^{-10}$$

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

$$8^{1}\times \frac{1}{14}q^{4-10}$$

Add the exponents 4 and -10.

$$8^{1}\times \frac{1}{14}q^{-6}$$

Raise 8 to the power 1.

$$8\times \frac{1}{14}q^{-6}$$

Multiply 8 times \frac{1}{14}.

$$\frac{4}{7}q^{-6}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{8^{1}q^{4}}{14^{1}q^{10}}$$

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

$$\frac{8^{1}q^{4-10}}{14^{1}}$$

Subtract 10 from 4.

$$\frac{8^{1}q^{-6}}{14^{1}}$$

Reduce the fraction \frac{8}{14} to lowest terms by extracting and canceling out 2.

$$\frac{4}{7}q^{-6}$$

Differentiate w.r.t. q

$$-\frac{24}{7q^{7}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}q}(\frac{8}{14}q^{4-10})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}q}(\frac{4}{7}q^{-6})$$

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}.

$$-6\times \frac{4}{7}q^{-6-1}$$

Do the arithmetic.

$$-\frac{24}{7}q^{-7}$$