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]
