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The given expression is:

\[
\frac{64a^2b^5}{35b^2c^4} \div \frac{12a^4b^3c}{70abc^8}
\]

Let's break down and simplify the expression step by step:

### Step 1: Rewrite the division as multiplication by the reciprocal
\[
\frac{64a^2b^5}{35b^2c^4} \times \frac{70abc^8}{12a^4b^3c}
\]

### Step 2: Multiply the fractions
Now, multiply the numerators and the denominators:

\[
\frac{64a^2b^5 \cdot 70abc^8}{35b^2c^4 \cdot 12a^4b^3c}
\]

### Step 3: Combine like terms
Multiply the constants:
\[
64 \times 70 = 4480
\]
\[
35 \times 12 = 420
\]

Now, simplify the powers of \( a \), \( b \), and \( c \):
- For \( a \): \( a^2 \times a = a^3 \) in the numerator, and \( a^4 \) in the denominator. So, we have \( a^{3-4} = a^{-1} \) or \( \frac{1}{a} \).
- For \( b \): \( b^5 \times b = b^6 \) in the numerator, and \( b^2 \times b^3 = b^5 \) in the denominator. So, the \( b \)'s cancel out.
- For \( c \): \( c^8 \) in the numerator, and \( c^4 \times c = c^5 \) in the denominator. So, we have \( c^{8-5} = c^3 \).

### Step 4: Simplify the expression
Now, the expression simplifies to:

\[
\frac{4480}{420} \times \frac{c^3}{a}
\]

Simplifying the constants:
\[
\frac{4480}{420} = \frac{448}{42} = \frac{224}{21}
\]

Thus, the simplified expression is:

\[
\frac{224c^3}{21a}
\]

### Final answer:
\[
\boxed{\frac{224c^3}{21a}}
\]

Would you like any further details or clarifications?

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Here are 5 related questions to deepen your understanding:

1. How do you simplify algebraic fractions involving exponents?
2. How would you handle different bases in the numerator and denominator of a fraction?
3. Can this process be applied to more complex rational expressions?
4. What rules apply when dividing powers of variables with the same base?
5. How can you interpret the physical meaning of a variable in an equation like this one?

**Tip:** When dividing fractions, always remember to multiply by the reciprocal, and simplify step by step to avoid errors.

Simplify the algebraic fraction: (64a^2b^5) / (35b^2c^4) ÷ (12a^4b^3c) / (70abc^8)

Learn how to simplify the algebraic expression (64a^2b^5) / (35b^2c^4) ÷ (12a^4b^3c) / (70abc^8) by following a detailed step-by-step solution. This problem involves multiplying rational expressions and applying the laws of exponents. Ideal for students in Grades 9-12.

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