Question

simplify.
\frac{v^{\frac{4}{5}}}{v^{\frac{2}{3}}}
write your answer using only a positive exponent.
assume that the variable represents a positive real number.

Explanation:

Step1: Use the quotient rule for exponents

When dividing two exponential expressions with the same base, we subtract the exponents: \( \frac{a^m}{a^n} = a^{m - n} \). Here, the base is \( v \), so we have \( v^{\frac{4}{5}} \div v^{\frac{2}{3}} = v^{\frac{4}{5} - \frac{2}{3}} \).

Step2: Subtract the fractions

Find a common denominator for \( \frac{4}{5} \) and \( \frac{2}{3} \), which is 15. Convert the fractions: \( \frac{4}{5} = \frac{12}{15} \) and \( \frac{2}{3} = \frac{10}{15} \). Then subtract: \( \frac{12}{15} - \frac{10}{15} = \frac{2}{15} \). So the exponent becomes \( \frac{2}{15} \), and the expression simplifies to \( v^{\frac{2}{15}} \).

Answer:

\( v^{\frac{2}{15}} \)