Let's go through the process of simplifying the expressions step by step.

Simplify the Following Exponential

1. $x^4 \cdot x^3$

   Using the law of exponents:
   $x^m \cdot x^n = x^{m+n}$
   So:
   $x^4 \cdot x^3 = x^{4+3} = x^7$

2. $(14x^2 y^3)^2$

   Applying the power of a product rule:
   $(ab)^n = a^n b^n$
   So:
   $(14x^2 y^3)^2 = 14^2 \cdot (x^2)^2 \cdot (y^3)^2 = 196x^4 y^6$

3. $\frac{12x^3}{6x^2}$

   Simplify the coefficients and subtract exponents:
   $\frac{12}{6} = 2, \quad x^3 / x^2 = x^{3-2} = x$
   So:
   $\frac{12x^3}{6x^2} = 2x$

4. $x^{-2} \cdot x^3$

   Using the exponent addition rule:
   $x^{-2} \cdot x^3 = x^{-2+3} = x^1 = x$

5. $(5n^2 n^4)^0$

   Anything raised to the power of zero equals 1:
   $(5n^2 n^4)^0 = 1$

6. $\left( \frac{2x^3}{y^{-2}} \right)^{-2}$

   First, apply the negative exponent rule:
   $\left( \frac{a}{b} \right)^{-n} = \frac{b^n}{a^n}$
   Then simplify the result: $\left( \frac{2x^3}{y^{-2}} \right)^{-2} = \frac{y^{4}}{4x^6}$

7. $3x^2 y^3 \cdot (-4xy^3)$

   Multiply coefficients and add exponents: $3 \cdot (-4) = -12, \quad x^2 \cdot x^1 = x^3, \quad y^3 \cdot y^3 = y^6$
   So:
   $3x^2 y^3 \cdot (-4xy^3) = -12x^3 y^6$

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