Solve 8g/3g^4

$$\frac { 8 g } { 3 g ^ { 4 } }$$

Evaluate

$$\frac{8}{3g^{3}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(8g^{1}\right)^{1}\times \frac{1}{3g^{4}}$$

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

$$8^{1}\left(g^{1}\right)^{1}\times \frac{1}{3}\times \frac{1}{g^{4}}$$

Use the Commutative Property of Multiplication.

$$8^{1}\times \frac{1}{3}\left(g^{1}\right)^{1}\times \frac{1}{g^{4}}$$

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

$$8^{1}\times \frac{1}{3}g^{1}g^{4\left(-1\right)}$$

Multiply 4 times -1.

$$8^{1}\times \frac{1}{3}g^{1}g^{-4}$$

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

$$8^{1}\times \frac{1}{3}g^{1-4}$$

Add the exponents 1 and -4.

$$8^{1}\times \frac{1}{3}g^{-3}$$

Raise 8 to the power 1.

$$8\times \frac{1}{3}g^{-3}$$

Multiply 8 times \frac{1}{3}.

$$\frac{8}{3}g^{-3}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{8^{1}g^{1}}{3^{1}g^{4}}$$

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

$$\frac{8^{1}g^{1-4}}{3^{1}}$$

Subtract 4 from 1.

$$\frac{8^{1}g^{-3}}{3^{1}}$$

Divide 8 by 3.

$$\frac{8}{3}g^{-3}$$

Differentiate w.r.t. g

$$-\frac{8}{g^{4}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}g}(\frac{8}{3}g^{1-4})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}g}(\frac{8}{3}g^{-3})$$

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}.

$$-3\times \frac{8}{3}g^{-3-1}$$

Do the arithmetic.

$$-8g^{-4}$$