Solve 40x^5/48x^6

$$\frac{ 40 { x }^{ 5 } }{ 48 { x }^{ 6 } }$$

Evaluate

$$\frac{5}{6x}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(40x^{5}\right)^{1}\times \frac{1}{48x^{6}}$$

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

$$40^{1}\left(x^{5}\right)^{1}\times \frac{1}{48}\times \frac{1}{x^{6}}$$

Use the Commutative Property of Multiplication.

$$40^{1}\times \frac{1}{48}\left(x^{5}\right)^{1}\times \frac{1}{x^{6}}$$

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

$$40^{1}\times \frac{1}{48}x^{5}x^{6\left(-1\right)}$$

Multiply 6 times -1.

$$40^{1}\times \frac{1}{48}x^{5}x^{-6}$$

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

$$40^{1}\times \frac{1}{48}x^{5-6}$$

Add the exponents 5 and -6.

$$40^{1}\times \frac{1}{48}\times \frac{1}{x}$$

Raise 40 to the power 1.

$$40\times \frac{1}{48}\times \frac{1}{x}$$

Multiply 40 times \frac{1}{48}.

$$\frac{5}{6}\times \frac{1}{x}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{40^{1}x^{5}}{48^{1}x^{6}}$$

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

$$\frac{40^{1}x^{5-6}}{48^{1}}$$

Subtract 6 from 5.

$$\frac{40^{1}\times \frac{1}{x}}{48^{1}}$$

Reduce the fraction \frac{40}{48} to lowest terms by extracting and canceling out 8.

$$\frac{5}{6}\times \frac{1}{x}$$

Differentiate w.r.t. x

$$-\frac{5}{6x^{2}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{40}{48}x^{5-6})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5}{6}\times \frac{1}{x})$$

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}.

$$-\frac{5}{6}x^{-1-1}$$

Do the arithmetic.

$$-\frac{5}{6}x^{-2}$$