Solve 21m^2/7m^3

$$\frac { 21 m ^ { 2 } } { 7 m ^ { 3 } }$$

Evaluate

$$\frac{3}{m}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(21m^{2}\right)^{1}\times \frac{1}{7m^{3}}$$

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

$$21^{1}\left(m^{2}\right)^{1}\times \frac{1}{7}\times \frac{1}{m^{3}}$$

Use the Commutative Property of Multiplication.

$$21^{1}\times \frac{1}{7}\left(m^{2}\right)^{1}\times \frac{1}{m^{3}}$$

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

$$21^{1}\times \frac{1}{7}m^{2}m^{3\left(-1\right)}$$

Multiply 3 times -1.

$$21^{1}\times \frac{1}{7}m^{2}m^{-3}$$

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

$$21^{1}\times \frac{1}{7}m^{2-3}$$

Add the exponents 2 and -3.

$$21^{1}\times \frac{1}{7}\times \frac{1}{m}$$

Raise 21 to the power 1.

$$21\times \frac{1}{7}\times \frac{1}{m}$$

Multiply 21 times \frac{1}{7}.

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

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{21^{1}m^{2}}{7^{1}m^{3}}$$

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

$$\frac{21^{1}m^{2-3}}{7^{1}}$$

Subtract 3 from 2.

$$\frac{21^{1}\times \frac{1}{m}}{7^{1}}$$

Divide 21 by 7.

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

Differentiate w.r.t. m

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

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{21}{7}m^{2-3})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}m}(3\times \frac{1}{m})$$

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}.

$$-3m^{-1-1}$$

Do the arithmetic.

$$-3m^{-2}$$