Solve 16m^2/24m^7

$$\frac{ 16 { m }^{ 2 } }{ 24 { m }^{ 7 } }$$

Evaluate

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

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(16m^{2}\right)^{1}\times \frac{1}{24m^{7}}$$

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

$$16^{1}\left(m^{2}\right)^{1}\times \frac{1}{24}\times \frac{1}{m^{7}}$$

Use the Commutative Property of Multiplication.

$$16^{1}\times \frac{1}{24}\left(m^{2}\right)^{1}\times \frac{1}{m^{7}}$$

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

$$16^{1}\times \frac{1}{24}m^{2}m^{7\left(-1\right)}$$

Multiply 7 times -1.

$$16^{1}\times \frac{1}{24}m^{2}m^{-7}$$

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

$$16^{1}\times \frac{1}{24}m^{2-7}$$

Add the exponents 2 and -7.

$$16^{1}\times \frac{1}{24}m^{-5}$$

Raise 16 to the power 1.

$$16\times \frac{1}{24}m^{-5}$$

Multiply 16 times \frac{1}{24}.

$$\frac{2}{3}m^{-5}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{16^{1}m^{2}}{24^{1}m^{7}}$$

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

$$\frac{16^{1}m^{2-7}}{24^{1}}$$

Subtract 7 from 2.

$$\frac{16^{1}m^{-5}}{24^{1}}$$

Reduce the fraction \frac{16}{24} to lowest terms by extracting and canceling out 8.

$$\frac{2}{3}m^{-5}$$

Differentiate w.r.t. m

$$-\frac{10}{3m^{6}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{16}{24}m^{2-7})$$

Do the arithmetic.

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

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}.

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

Do the arithmetic.

$$-\frac{10}{3}m^{-6}$$