Solve 40v^3/24v^2

$$\frac { 40 v ^ { 3 } } { 24 v ^ { 2 } }$$

Evaluate

$$\frac{5v}{3}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(40v^{3}\right)^{1}\times \frac{1}{24v^{2}}$$

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

$$40^{1}\left(v^{3}\right)^{1}\times \frac{1}{24}\times \frac{1}{v^{2}}$$

Use the Commutative Property of Multiplication.

$$40^{1}\times \frac{1}{24}\left(v^{3}\right)^{1}\times \frac{1}{v^{2}}$$

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

$$40^{1}\times \frac{1}{24}v^{3}v^{2\left(-1\right)}$$

Multiply 2 times -1.

$$40^{1}\times \frac{1}{24}v^{3}v^{-2}$$

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

$$40^{1}\times \frac{1}{24}v^{3-2}$$

Add the exponents 3 and -2.

$$40^{1}\times \frac{1}{24}v^{1}$$

Raise 40 to the power 1.

$$40\times \frac{1}{24}v^{1}$$

Multiply 40 times \frac{1}{24}.

$$\frac{5}{3}v^{1}$$

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

$$\frac{5}{3}v$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{40^{1}v^{3}}{24^{1}v^{2}}$$

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

$$\frac{40^{1}v^{3-2}}{24^{1}}$$

Subtract 2 from 3.

$$\frac{40^{1}v^{1}}{24^{1}}$$

Reduce the fraction \frac{40}{24} to lowest terms by extracting and canceling out 8.

$$\frac{5}{3}v^{1}$$

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

$$\frac{5}{3}v$$

Differentiate w.r.t. v

$$\frac{5}{3} = 1\frac{2}{3} = 1.6666666666666667$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}v}(\frac{40}{24}v^{3-2})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}v}(\frac{5}{3}v^{1})$$

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}.

$$\frac{5}{3}v^{1-1}$$

Do the arithmetic.

$$\frac{5}{3}v^{0}$$

For any term t except 0, t^{0}=1.

$$\frac{5}{3}\times 1$$

For any term t, t\times 1=t and 1t=t.

$$\frac{5}{3}$$