Solve -3b^-2/(6b^-7)

$$\frac { - 3 b ^ { - 2 } } { ( 6 b ^ { - 7 } ) }$$

Evaluate

$$-\frac{b^{5}}{2}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(-3b^{-2}\right)^{1}\times \frac{1}{6b^{-7}}$$

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

$$\left(-3\right)^{1}\left(b^{-2}\right)^{1}\times \frac{1}{6}\times \frac{1}{b^{-7}}$$

Use the Commutative Property of Multiplication.

$$\left(-3\right)^{1}\times \frac{1}{6}\left(b^{-2}\right)^{1}\times \frac{1}{b^{-7}}$$

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

$$\left(-3\right)^{1}\times \frac{1}{6}b^{-2}b^{-7\left(-1\right)}$$

Multiply -7 times -1.

$$\left(-3\right)^{1}\times \frac{1}{6}b^{-2}b^{7}$$

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

$$\left(-3\right)^{1}\times \frac{1}{6}b^{-2+7}$$

Add the exponents -2 and 7.

$$\left(-3\right)^{1}\times \frac{1}{6}b^{5}$$

Raise -3 to the power 1.

$$-3\times \frac{1}{6}b^{5}$$

Multiply -3 times \frac{1}{6}.

$$-\frac{1}{2}b^{5}$$

Differentiate w.r.t. b

$$-\frac{5b^{4}}{2}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}b}(\left(-\frac{3}{6}\right)b^{-2-\left(-7\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}b}(-\frac{1}{2}b^{5})$$

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}.

$$5\left(-\frac{1}{2}\right)b^{5-1}$$

Do the arithmetic.

$$-\frac{5}{2}b^{4}$$