Solve 54v^4/9v^5

$$\frac { 54 v ^ { 4 } } { 9 v ^ { 5 } }$$

Evaluate

$$\frac{6}{v}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(54v^{4}\right)^{1}\times \frac{1}{9v^{5}}$$

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

$$54^{1}\left(v^{4}\right)^{1}\times \frac{1}{9}\times \frac{1}{v^{5}}$$

Use the Commutative Property of Multiplication.

$$54^{1}\times \frac{1}{9}\left(v^{4}\right)^{1}\times \frac{1}{v^{5}}$$

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

$$54^{1}\times \frac{1}{9}v^{4}v^{5\left(-1\right)}$$

Multiply 5 times -1.

$$54^{1}\times \frac{1}{9}v^{4}v^{-5}$$

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

$$54^{1}\times \frac{1}{9}v^{4-5}$$

Add the exponents 4 and -5.

$$54^{1}\times \frac{1}{9}\times \frac{1}{v}$$

Raise 54 to the power 1.

$$54\times \frac{1}{9}\times \frac{1}{v}$$

Multiply 54 times \frac{1}{9}.

$$6\times \frac{1}{v}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{54^{1}v^{4}}{9^{1}v^{5}}$$

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

$$\frac{54^{1}v^{4-5}}{9^{1}}$$

Subtract 5 from 4.

$$\frac{54^{1}\times \frac{1}{v}}{9^{1}}$$

Divide 54 by 9.

$$6\times \frac{1}{v}$$

Differentiate w.r.t. v

$$-\frac{6}{v^{2}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}v}(\frac{54}{9}v^{4-5})$$

Do the arithmetic.

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

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}.

$$-6v^{-1-1}$$

Do the arithmetic.

$$-6v^{-2}$$