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: Simplify the coefficient

First, simplify the fraction of the coefficients. The coefficient of the numerator is 4 and the coefficient of the denominator is 2. So, $\frac{4}{2} = 2$.

Step2: Simplify the variable with exponents

For the variable part, we use the quotient rule of exponents, which states that $\frac{a^m}{a^n}=a^{m - n}$ when $a
eq0$. Here, $a = v$, $m=\frac{4}{5}$, and $n=\frac{2}{3}$. So we calculate the exponent: $\frac{4}{5}-\frac{2}{3}$. To subtract these fractions, find a common denominator, which is 15. So $\frac{4}{5}=\frac{12}{15}$ and $\frac{2}{3}=\frac{10}{15}$. Then $\frac{12}{15}-\frac{10}{15}=\frac{2}{15}$. So the variable part simplifies to $v^{\frac{2}{15}}$.

Step3: Combine the coefficient and the variable

Now, combine the simplified coefficient and the simplified variable part. We have $2\times v^{\frac{2}{15}} = 2v^{\frac{2}{15}}$.

Answer:

$2v^{\frac{2}{15}}$