Solve 54m^4/6m^6

$$\frac { 54 m ^ { 4 } } { 6 m ^ { 6 } }$$

Evaluate

$$\frac{9}{m^{2}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(54m^{4}\right)^{1}\times \frac{1}{6m^{6}}$$

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

$$54^{1}\left(m^{4}\right)^{1}\times \frac{1}{6}\times \frac{1}{m^{6}}$$

Use the Commutative Property of Multiplication.

$$54^{1}\times \frac{1}{6}\left(m^{4}\right)^{1}\times \frac{1}{m^{6}}$$

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

$$54^{1}\times \frac{1}{6}m^{4}m^{6\left(-1\right)}$$

Multiply 6 times -1.

$$54^{1}\times \frac{1}{6}m^{4}m^{-6}$$

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

$$54^{1}\times \frac{1}{6}m^{4-6}$$

Add the exponents 4 and -6.

$$54^{1}\times \frac{1}{6}m^{-2}$$

Raise 54 to the power 1.

$$54\times \frac{1}{6}m^{-2}$$

Multiply 54 times \frac{1}{6}.

$$9m^{-2}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{54^{1}m^{4}}{6^{1}m^{6}}$$

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

$$\frac{54^{1}m^{4-6}}{6^{1}}$$

Subtract 6 from 4.

$$\frac{54^{1}m^{-2}}{6^{1}}$$

Divide 54 by 6.

$$9m^{-2}$$

Differentiate w.r.t. m

$$-\frac{18}{m^{3}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{54}{6}m^{4-6})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}m}(9m^{-2})$$

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}.

$$-2\times 9m^{-2-1}$$

Do the arithmetic.

$$-18m^{-3}$$