To simplify the expression
[tex]\[
\frac{v^{-\frac{1}{3}}}{v^{-\frac{5}{7}} v^{-\frac{2}{7}}}
\][/tex]
and write it using only positive exponents, follow these steps:
1. Combine the exponents in the denominator:
[tex]\[
v^{-\frac{5}{7}} v^{-\frac{2}{7}} = v^{-\frac{5}{7} - \frac{2}{7}}
\][/tex]
Add the exponents in the denominator:
[tex]\[
-\frac{5}{7} - \frac{2}{7} = -\frac{7}{7} = -1
\][/tex]
So, the denominator becomes:
[tex]\[
v^{-1}
\][/tex]
2. Combine the exponents in the fraction:
Now, we have:
[tex]\[
\frac{v^{-\frac{1}{3}}}{v^{-1}}
\][/tex]
This can be simplified by subtracting the exponent of the denominator from the exponent of the numerator:
[tex]\[
v^{-\frac{1}{3} - (-1)} = v^{-\frac{1}{3} + 1}
\][/tex]
3. Simplify the exponent:
[tex]\[
-\frac{1}{3} + 1 = -\frac{1}{3} + \frac{3}{3} = \frac{2}{3}
\][/tex]
So, the expression simplifies to:
[tex]\[
v^{\frac{2}{3}}
\][/tex]
Thus, the simplified expression with positive exponents is:
[tex]\[
v^{\frac{2}{3}}
\][/tex]
So the answer is:
[tex]\[
v^{\frac{2}{3}}
\][/tex]
