To simplify the given expression, we need to apply the properties of exponents. Specifically, we will:
- Simplify the exponent in the numerator by multiplying the exponents inside the parentheses.
- Simplify the resulting expression by dividing the exponents of \(a\) in the numerator and the denominator.
We start with the expression
\[
\frac{\left(a^{7 / 5} b^{4 / 3}\right)^{5 / 14}}{a^{7 / 6}}.
\]
First, we simplify the numerator by applying the exponent to both \(a\) and \(b\):
\[
\left(a^{7/5}\right)^{5/14} = a^{(7/5) \cdot (5/14)} = a^{1/2}
\]
and
\[
\left(b^{4/3}\right)^{5/14} = b^{(4/3) \cdot (5/14)} = b^{20/42} = b^{10/21}.
\]
Thus, the numerator simplifies to
\[
a^{1/2} b^{10/21}.
\]
Now, we rewrite the entire expression:
\[
\frac{a^{1/2} b^{10/21}}{a^{7/6}}.
\]
Next, we simplify the exponent of \(a\) by subtracting the exponent in the denominator from the exponent in the numerator:
\[
1/2 - 7/6 = \frac{3}{6} - \frac{7}{6} = -\frac{4}{6} = -\frac{2}{3}.
\]
Thus, the expression becomes:
\[
a^{-2/3} b^{10/21}.
\]
Now we substitute \(a = 2\) and \(b = 3\) into the simplified expression:
\[
a^{-2/3} b^{10/21} = 2^{-2/3} \cdot 3^{10/21}.
\]
Calculating each term, we find:
\[
2^{-2/3} = \frac{1}{\sqrt[3]{4}} \quad \text{and} \quad 3^{10/21} \approx 1.062952686026326.
\]
Thus, the final result is approximately:
\[
\frac{1}{\sqrt[3]{4}} \cdot 1.062952686026326 \approx 1.0629.
\]
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