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Let's simplify the given expression step by step. The expression provided is:

\[
\frac{4a^6b^7c^{12}}{12a^5b^8c^{13}}
\]

### Step 1: Simplifying the constants
We have \( \frac{4}{12} \), which simplifies to \( \frac{1}{3} \).

### Step 2: Simplifying the powers of \(a\)
Using the law of exponents \( \frac{a^m}{a^n} = a^{m-n} \):
\[
\frac{a^6}{a^5} = a^{6-5} = a^1 = a
\]

### Step 3: Simplifying the powers of \(b\)
Similarly, for \(b\):
\[
\frac{b^7}{b^8} = b^{7-8} = b^{-1} = \frac{1}{b}
\]

### Step 4: Simplifying the powers of \(c\)
For \(c\):
\[
\frac{c^{12}}{c^{13}} = c^{12-13} = c^{-1} = \frac{1}{c}
\]

### Step 5: Putting it all together
Now we can combine everything:
\[
\frac{4a^6b^7c^{12}}{12a^5b^8c^{13}} = \frac{1}{3} \cdot a \cdot \frac{1}{b} \cdot \frac{1}{c} = \frac{a}{3bc}
\]

Thus, the simplified expression is:

\[
\frac{a}{3bc}
\]

Would you like further details or have any questions?

### Related Questions:
1. How do exponent rules apply to division and multiplication?
2. Can negative exponents always be simplified to fractions?
3. How do you simplify expressions involving variables and constants?
4. What are the other laws of exponents, and how are they used?
5. What is the significance of fractional coefficients in algebraic expressions?

### Tip:
Always simplify constants first before applying exponent rules to variables.

Simplify the expression: (4a^6b^7c^12) / (12a^5b^8c^13).

Learn how to simplify the algebraic expression (4a^6b^7c^12) / (12a^5b^8c^13) using exponent rules. This step-by-step guide is suitable for students in Grades 8-10, covering exponent division and constant simplification.

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