Solve (21m^3/3m^2)

$$( \frac { 21 m ^ { 3 } } { 3 m ^ { 2 } } )$$

Evaluate

$$7m$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

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

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

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

Use the Commutative Property of Multiplication.

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

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

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

Multiply 2 times -1.

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

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

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

Add the exponents 3 and -2.

$$21^{1}\times \frac{1}{3}m^{1}$$

Raise 21 to the power 1.

$$21\times \frac{1}{3}m^{1}$$

Multiply 21 times \frac{1}{3}.

$$7m^{1}$$

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

$$7m$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

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

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

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

Subtract 2 from 3.

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

Divide 21 by 3.

$$7m^{1}$$

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

$$7m$$

Differentiate w.r.t. m

$$7$$

Solution Steps

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

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

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}m}(7m^{1})$$

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}.

$$7m^{1-1}$$

Do the arithmetic.

$$7m^{0}$$

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

$$7\times 1$$

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

$$7$$