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.