\frac{\left(\frac{3}{7}\right)^{-3}\cdot \left(\frac{a}{2}\right)^{-4}}{\left(-\frac{1}{7}\right)^{-3}\cdot \left(\frac{2}{a}\right)^{-2}}

We are given the expression:

\[
\frac{\left(\frac{3}{7}\right)^{-3} \cdot \left(\frac{a}{2}\right)^{-4}}{\left(-\frac{1}{7}\right)^{-3} \cdot \left(\frac{2}{a}\right)^{-2}}
\]

To simplify this, we will break it down step by step.

### Step 1: Apply the Negative Exponent Rule
The negative exponent rule states that:
\[
\left(\frac{x}{y}\right)^{-n} = \left(\frac{y}{x}\right)^n
\]

Using this rule, we can simplify the terms:

\[
\left(\frac{3}{7}\right)^{-3} = \left(\frac{7}{3}\right)^3 = \frac{7^3}{3^3} = \frac{343}{27}
\]

\[
\left(\frac{a}{2}\right)^{-4} = \left(\frac{2}{a}\right)^4 = \frac{2^4}{a^4} = \frac{16}{a^4}
\]

\[
\left(-\frac{1}{7}\right)^{-3} = \left(-\frac{7}{1}\right)^3 = (-7)^3 = -343
\]

\[
\left(\frac{2}{a}\right)^{-2} = \left(\frac{a}{2}\right)^2 = \frac{a^2}{2^2} = \frac{a^2}{4}
\]

### Step 2: Substitute the simplified terms
Now, we can substitute the simplified terms back into the original expression:

\[
\frac{\frac{343}{27} \cdot \frac{16}{a^4}}{-343 \cdot \frac{a^2}{4}}
\]

### Step 3: Simplify the expression
First, simplify the numerator and denominator separately:

#### Numerator:
\[
\frac{343}{27} \cdot \frac{16}{a^4} = \frac{343 \cdot 16}{27 \cdot a^4} = \frac{5488}{27a^4}
\]

#### Denominator:
\[
-343 \cdot \frac{a^2}{4} = \frac{-343a^2}{4}
\]

Now, rewrite the overall expression:

\[
\frac{\frac{5488}{27a^4}}{\frac{-343a^2}{4}} = \frac{5488}{27a^4} \cdot \frac{4}{-343a^2}
\]

### Step 4: Multiply and simplify
Now, multiply the fractions:

\[
\frac{5488 \cdot 4}{27 \cdot (-343) \cdot a^4 \cdot a^2} = \frac{21952}{-9261 a^6}
\]

Thus, the simplified form is:

\[
\boxed{-\frac{21952}{9261 a^6}}
\]

### Final answer:
\[
-\frac{21952}{9261 a^6}
\]

Would you like more details on any step or further clarification?

---

Here are five related questions for practice:

1. Simplify: \(\left(\frac{5}{6}\right)^{-2} \cdot \left(\frac{a}{3}\right)^3\)
2. Evaluate: \(\frac{\left(\frac{2}{5}\right)^{-4}}{\left(\frac{3}{7}\right)^{-2}}\)
3. Simplify: \(\left(\frac{b}{c}\right)^{-3} \cdot \left(\frac{c}{d}\right)^{2}\)
4. Solve: \(\left(\frac{a}{b}\right)^{-3} = \frac{125}{64}\), find \(a\) and \(b\).
5. Simplify: \(\frac{(x^{-2}y^3)^2}{(x^3y^{-4})^{-1}}\)

### Tip: When simplifying expressions with exponents, always apply exponent rules carefully, especially when dealing with negative exponents.

\frac{\left(\frac{3}{7}\right)^{-3} \cdot \left(\frac{a}{2}\right)^{-4}}{\left(-\frac{1}{7}\right)^{-3} \cdot \left(\frac{2}{a}\right)^{-2}}

Learn how to simplify an expression involving negative exponents and fractional bases: \frac{\left(\frac{3}{7}\right)^{-3} \cdot \left(\frac{a}{2}\right)^{-4}}{\left(-\frac{1}{7}\right)^{-3} \cdot \left(\frac{2}{a}\right)^{-2}}. This step-by-step solution demonstrates key exponent rules and fraction multiplication concepts. Suitable for high school students in Grades 9-12.

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