Certainly! Let's simplify the given expression step-by-step:
The original expression to simplify is:
[tex]\[
\frac{v^{-\frac{1}{3}}}{v^{-\frac{6}{7}} v^{-\frac{2}{7}}}
\][/tex]
First, combine the exponents in the denominator:
[tex]\[
v^{-\frac{6}{7}} \cdot v^{-\frac{2}{7}} = v^{-\frac{6}{7} - \frac{2}{7}} = v^{-\frac{8}{7}}
\][/tex]
Now, the expression simplifies to:
[tex]\[
\frac{v^{-\frac{1}{3}}}{v^{-\frac{8}{7}}}
\][/tex]
Using the property of exponents that states [tex]\( \frac{a^m}{a^n} = a^{m-n} \)[/tex], we get:
[tex]\[
v^{-\frac{1}{3}} \div v^{-\frac{8}{7}} = v^{-\frac{1}{3} - (-\frac{8}{7})} = v^{-\frac{1}{3} + \frac{8}{7}}
\][/tex]
To combine the exponents [tex]\(-\frac{1}{3}\)[/tex] and [tex]\(\frac{8}{7}\)[/tex], we need a common denominator. The common denominator for 3 and 7 is 21.
Converting:
[tex]\[
-\frac{1}{3} = -\frac{7}{21}
\][/tex]
[tex]\[
\frac{8}{7} = \frac{24}{21}
\][/tex]
Now, add the exponents:
[tex]\[
-\frac{7}{21} + \frac{24}{21} = \frac{-7 + 24}{21} = \frac{17}{21}
\][/tex]
Thus, the expression simplifies to:
[tex]\[
v^{\frac{17}{21}}
\][/tex]
So, the simplified expression using only positive exponents is:
[tex]\[
v^{\frac{17}{21}}
\][/tex]
