To simplify the expression [tex]\( 8y^3 w \cdot 5v^5 v^{-1} \cdot 2w^5 y^{-4} \)[/tex], follow these steps:
1. Combine the coefficients: Multiply the numerical coefficients together:
[tex]\[
8 \cdot 5 \cdot 2
\][/tex]
Calculating this, we get:
[tex]\[
8 \cdot 5 = 40
\][/tex]
[tex]\[
40 \cdot 2 = 80
\][/tex]
2. Combine the exponents for [tex]\(y\)[/tex]: Add the exponents of [tex]\(y\)[/tex]:
[tex]\[
y^3 \text{ and } y^{-4}
\][/tex]
Using the property of exponents [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]:
[tex]\[
y^{3 + (-4)} = y^{-1}
\][/tex]
3. Combine the exponents for [tex]\(w\)[/tex]: Add the exponents of [tex]\(w\)[/tex]:
[tex]\[
w^1 \text{ and } w^5
\][/tex]
Using the property of exponents [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]:
[tex]\[
w^{1 + 5} = w^6
\][/tex]
4. Combine the exponents for [tex]\(v\)[/tex]: Add the exponents of [tex]\(v\)[/tex]:
[tex]\[
v^5 \text{ and } v^{-1}
\][/tex]
Using the property of exponents [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]:
[tex]\[
v^{5 + (-1)} = v^4
\][/tex]
5. Construct the simplified expression: Now combine all the parts together:
[tex]\[
80 \cdot y^{-1} \cdot w^6 \cdot v^4
\][/tex]
6. Rewrite with positive exponents: Since we want the final answer with only positive exponents, remember that [tex]\(y^{-1} = \frac{1}{y}\)[/tex]:
[tex]\[
80 v^4 w^6 \cdot \frac{1}{y} = \frac{80 v^4 w^6}{y}
\][/tex]
Therefore, the simplified expression is:
[tex]\[
\frac{80 v^4 w^6}{y}
\][/tex]
