We are given the expression:

$x^{-2} \times x^7 \times x^{4x-2} \times x^7 \times x^4$

To simplify, we need to apply the product of powers rule, which states:

$x^a \times x^b = x^{a + b}$

Now, let's break it down step by step:

1. First, combine the exponents of the same base (all terms have base $x$): $x^{-2} \times x^7 = x^{-2 + 7} = x^5$ $x^5 \times x^{4x-2} = x^{5 + (4x-2)} = x^{4x + 3}$ $x^{4x+3} \times x^7 = x^{(4x + 3) + 7} = x^{4x + 10}$ $x^{4x+10} \times x^4 = x^{(4x + 10) + 4} = x^{4x + 14}$

So, the simplified expression is:

$x^{4x + 14}$

Final Answer:

$x^{4x + 14}$

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Would you like further clarification on any of the steps? Or have any questions?

Here are 5 questions related to this topic:

1. What is the rule for multiplying terms with the same base in exponents?
2. How do you handle expressions with variables in the exponents, like $4x-2$?
3. What happens if there are negative exponents in a multiplication of terms?
4. How do you simplify a product of multiple exponential terms?
5. What is the relationship between powers of the same base when dividing?

Tip: Always simplify exponents step-by-step when multiplying, and be careful with signs, especially with negative exponents!