Solve 4b^8/8b^-3

$$\frac { 4 b ^ { 8 } } { 8 b ^ { - 3 } }$$

Evaluate

$$\frac{b^{11}}{2}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

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

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

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

Use the Commutative Property of Multiplication.

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

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

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

Multiply -3 times -1.

$$4^{1}\times \frac{1}{8}b^{8}b^{3}$$

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

$$4^{1}\times \frac{1}{8}b^{8+3}$$

Add the exponents 8 and 3.

$$4^{1}\times \frac{1}{8}b^{11}$$

Raise 4 to the power 1.

$$4\times \frac{1}{8}b^{11}$$

Multiply 4 times \frac{1}{8}.

$$\frac{1}{2}b^{11}$$

Differentiate w.r.t. b

$$\frac{11b^{10}}{2}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{4}{8}b^{8-\left(-3\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{1}{2}b^{11})$$

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}.

$$11\times \frac{1}{2}b^{11-1}$$

Do the arithmetic.

$$\frac{11}{2}b^{10}$$