Let's simplify the given expression:

\[
\frac{(-2)^3 \times (-2)^7}{3 \times 4^6}
\]

### Step 1: Combine powers of \((-2)\)
We use the property of exponents that states \( a^m \times a^n = a^{m+n} \). Therefore:

\[
(-2)^3 \times (-2)^7 = (-2)^{3+7} = (-2)^{10}
\]

Now the expression becomes:

\[
\frac{(-2)^{10}}{3 \times 4^6}
\]

### Step 2: Rewrite \( 4^6 \)
Since \( 4 = 2^2 \), we can rewrite \( 4^6 \) as:

\[
4^6 = (2^2)^6 = 2^{12}
\]

### Step 3: Simplify the expression
Now substitute this back into the equation:

\[
\frac{(-2)^{10}}{3 \times 2^{12}}
\]

Since \((-2)^{10} = 2^{10}\), the expression becomes:

\[
\frac{2^{10}}{3 \times 2^{12}}
\]

### Step 4: Apply the properties of exponents
Now, apply the property of exponents \(\frac{a^m}{a^n} = a^{m-n}\):

\[
\frac{2^{10}}{2^{12}} = 2^{10-12} = 2^{-2}
\]

Thus, the expression becomes:

\[
\frac{2^{-2}}{3} = \frac{1}{3 \times 2^2} = \frac{1}{3 \times 4} = \frac{1}{12}
\]

### Final Answer:
\[
\boxed{\frac{1}{12}}
\]

Would you like further details on any step or have any questions?

Here are 5 related questions:
1. What happens when you multiply powers with the same base but different exponents?
2. How do you simplify expressions involving negative exponents?
3. Can you express \( (-2)^5 \times (-2)^4 \) in terms of a single power of -2?
4. How does the base being negative impact the powers in \( (-2)^n \)?
5. Can you explain how \( 4^6 = 2^{12} \) was derived?

**Tip:** When simplifying expressions with exponents, always group similar bases together first before applying exponent rules.

Explore how to simplify the expression ((-2)^3 x (-2)^7) / 3 x 4^6 step-by-step using the laws of exponents. Understand key mathematical concepts such as combining exponents and simplifying expressions with powers of negative bases. Ideal for Grades 9-10 students.

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