Solve 4b^-1/b^-3

$$\frac { 4 b ^ { - 1 } } { b ^ { - 3 } }$$

Evaluate

$$4b^{2}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

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

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

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

Use the Commutative Property of Multiplication.

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

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

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

Multiply -3 times -1.

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

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

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

Add the exponents -1 and 3.

$$4^{1}\times \frac{1}{1}b^{2}$$

Raise 4 to the power 1.

$$4\times \frac{1}{1}b^{2}$$

Differentiate w.r.t. b

$$8b$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{4}{1}b^{-1-\left(-3\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}b}(4b^{2})$$

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}.

$$2\times 4b^{2-1}$$

Do the arithmetic.

$$8b^{1}$$

For any term t, t^{1}=t.

$$8b$$