Answer :

To simplify the expression [tex]\(7 v^{-1} \cdot 2 w^{-2} \cdot 4 u^6 w^4 v u^{-2}\)[/tex], we will follow these steps:

1. Combine the constants:
The constants given in the expression are 7, 2, and 4.
[tex]\[ 7 \cdot 2 \cdot 4 = 56 \][/tex]

So, we start with:
[tex]\[ 56 v^{-1} w^{-2} u^6 w^4 v u^{-2} \][/tex]

2. Combine the exponents of the same base:
- For [tex]\(v\)[/tex]:
The exponents are [tex]\(-1\)[/tex] and [tex]\(+1\)[/tex] (from [tex]\(v\)[/tex] and [tex]\(v^{-1}\)[/tex]).
[tex]\[ v^{-1 + 1} = v^0 \][/tex]
Since any base raised to the power of 0 is 1, [tex]\(v^0 = 1\)[/tex].

- For [tex]\(w\)[/tex]:
The exponents are [tex]\(-2\)[/tex] and [tex]\(+4\)[/tex] (from [tex]\(w^{-2}\)[/tex] and [tex]\(w^4\)[/tex]).
[tex]\[ w^{-2 + 4} = w^2 \][/tex]

- For [tex]\(u\)[/tex]:
The exponents are [tex]\(+6\)[/tex] and [tex]\(-2\)[/tex] (from [tex]\(u^6\)[/tex] and [tex]\(u^{-2}\)[/tex]).
[tex]\[ u^{6 - 2} = u^4 \][/tex]

3. Substitute the combined exponents back into the expression:
Since [tex]\(v^0 = 1\)[/tex], the [tex]\(v\)[/tex] terms cancel out:
[tex]\[ 56 w^2 u^4 \][/tex]

Therefore, the simplified expression is:
[tex]\[ 56 w^2 u^4 \][/tex]