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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:  
   \[
   3x^2 y^3 \cdot (-4xy^3) = -12x^3 y^6
   \]

8. **

Simplify the given exponential and radical expressions.

Explore the process of simplifying exponential and radical expressions with a detailed step-by-step breakdown. This guide covers essential algebraic concepts, including exponent rules, radical properties, and more. Suitable for students in Grades 8-10.

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