Solve 4r^-5/10r^-8

$$\frac { 4 r ^ { - 5 } } { 10 r ^ { - 8 } }$$

Evaluate

$$\frac{2r^{3}}{5}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(4r^{-5}\right)^{1}\times \frac{1}{10r^{-8}}$$

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

$$4^{1}\left(r^{-5}\right)^{1}\times \frac{1}{10}\times \frac{1}{r^{-8}}$$

Use the Commutative Property of Multiplication.

$$4^{1}\times \frac{1}{10}\left(r^{-5}\right)^{1}\times \frac{1}{r^{-8}}$$

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

$$4^{1}\times \frac{1}{10}r^{-5}r^{-8\left(-1\right)}$$

Multiply -8 times -1.

$$4^{1}\times \frac{1}{10}r^{-5}r^{8}$$

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

$$4^{1}\times \frac{1}{10}r^{-5+8}$$

Add the exponents -5 and 8.

$$4^{1}\times \frac{1}{10}r^{3}$$

Raise 4 to the power 1.

$$4\times \frac{1}{10}r^{3}$$

Multiply 4 times \frac{1}{10}.

$$\frac{2}{5}r^{3}$$

Differentiate w.r.t. r

$$\frac{6r^{2}}{5}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}r}(\frac{4}{10}r^{-5-\left(-8\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}r}(\frac{2}{5}r^{3})$$

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}.

$$3\times \frac{2}{5}r^{3-1}$$

Do the arithmetic.

$$\frac{6}{5}r^{2}$$