Solve 40u^4/5u^6

$$\frac { 40 u ^ { 4 } } { 5 u ^ { 6 } }$$

Evaluate

$$\frac{8}{u^{2}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(40u^{4}\right)^{1}\times \frac{1}{5u^{6}}$$

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

$$40^{1}\left(u^{4}\right)^{1}\times \frac{1}{5}\times \frac{1}{u^{6}}$$

Use the Commutative Property of Multiplication.

$$40^{1}\times \frac{1}{5}\left(u^{4}\right)^{1}\times \frac{1}{u^{6}}$$

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

$$40^{1}\times \frac{1}{5}u^{4}u^{6\left(-1\right)}$$

Multiply 6 times -1.

$$40^{1}\times \frac{1}{5}u^{4}u^{-6}$$

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

$$40^{1}\times \frac{1}{5}u^{4-6}$$

Add the exponents 4 and -6.

$$40^{1}\times \frac{1}{5}u^{-2}$$

Raise 40 to the power 1.

$$40\times \frac{1}{5}u^{-2}$$

Multiply 40 times \frac{1}{5}.

$$8u^{-2}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{40^{1}u^{4}}{5^{1}u^{6}}$$

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

$$\frac{40^{1}u^{4-6}}{5^{1}}$$

Subtract 6 from 4.

$$\frac{40^{1}u^{-2}}{5^{1}}$$

Divide 40 by 5.

$$8u^{-2}$$

Differentiate w.r.t. u

$$-\frac{16}{u^{3}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}u}(\frac{40}{5}u^{4-6})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}u}(8u^{-2})$$

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}.

$$-2\times 8u^{-2-1}$$

Do the arithmetic.

$$-16u^{-3}$$