Solve v^frac{7{3}}*v^frac{5{3}}*v^-frac{1{3}}

$$v ^ { \frac { 7 } { 3 } } \cdot v ^ { \frac { 5 } { 3 } } \cdot v ^ { - \frac { 1 } { 3 } }$$

Evaluate

$$v^{\frac{11}{3}}$$

Steps Using Product of Powers Property

Use the rules of exponents to simplify the expression.

$$v^{\frac{7}{3}}v^{\frac{5}{3}}v^{-\frac{1}{3}}$$

Use the Multiplication Rule for Exponents.

$$v^{\frac{7+5-1}{3}}$$

Add the exponents \frac{7}{3} and \frac{5}{3}.

$$v^{4-\frac{1}{3}}$$

Add the exponents 4 and -\frac{1}{3}.

$$v^{\frac{11}{3}}$$

Differentiate w.r.t. v

$$\frac{11v^{\frac{8}{3}}}{3}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add \frac{7}{3} and \frac{5}{3} to get 4.

$$\frac{\mathrm{d}}{\mathrm{d}v}(v^{4}v^{-\frac{1}{3}})$$

To multiply powers of the same base, add their exponents. Add 4 and -\frac{1}{3} to get \frac{11}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}v}(v^{\frac{11}{3}})$$

The derivative of ax^{n} is nax^{n-1}.

$$\frac{11}{3}v^{\frac{11}{3}-1}$$

Subtract 1 from \frac{11}{3}.

$$\frac{11}{3}v^{\frac{8}{3}}$$