To simplify the expression [tex]\(\frac{v^{\frac{1}{8}}}{v^{\frac{3}{7}}}\)[/tex], let's use the rules of exponents.
1. Apply the Property of Exponents:
When dividing like bases with exponents, we subtract the exponent in the denominator from the exponent in the numerator. The property can be stated as:
[tex]\[
\frac{a^m}{a^n} = a^{m-n}
\][/tex]
Here, our base is [tex]\(v\)[/tex], and the exponents are [tex]\(\frac{1}{8}\)[/tex] and [tex]\(\frac{3}{7}\)[/tex].
2. Subtract the Exponents:
We need to find the result of subtracting [tex]\(\frac{3}{7}\)[/tex] from [tex]\(\frac{1}{8}\)[/tex]. Mathematically, this can be expressed as:
[tex]\[
\frac{1}{8} - \frac{3}{7}
\][/tex]
Simplifying this expression involves finding a common denominator. However, we have already determined that:
[tex]\[
\frac{1}{8} - \frac{3}{7} = -0.30357142857142855
\][/tex]
3. Express the Result with a Positive Exponent:
The result of [tex]\(\frac{1}{8} - \frac{3}{7}\)[/tex] is a negative exponent. To write the final answer using a positive exponent, we take the absolute value of the result:
[tex]\[
\left| -0.30357142857142855 \right| = 0.30357142857142855
\][/tex]
Hence, the simplified form of the given expression, using a positive exponent, is:
[tex]\[
v^{0.30357142857142855}
\][/tex]
Thus, the answer is:
[tex]\[
v^{0.30357142857142855}
\][/tex]
