Let's simplify the given expression step-by-step:
[tex]\[
\frac{4 x^2 y^7}{32 x^4 y^3} \cdot \frac{16 x^2}{8 y^6}
\][/tex]
Step 1: Simplify the first fraction [tex]\(\frac{4 x^2 y^7}{32 x^4 y^3}\)[/tex]
- Simplify the coefficients:
[tex]\[
\frac{4}{32} = \frac{1}{8}
\][/tex]
- Simplify the [tex]\(x\)[/tex]-terms:
[tex]\[
\frac{x^2}{x^4} = x^{2-4} = x^{-2}
\][/tex]
- Simplify the [tex]\(y\)[/tex]-terms:
[tex]\[
\frac{y^7}{y^3} = y^{7-3} = y^4
\][/tex]
So, the first fraction simplifies to:
[tex]\[
\frac{1}{8} \cdot x^{-2} \cdot y^4 = \frac{y^4}{8 x^2}
\][/tex]
Step 2: Simplify the second fraction [tex]\(\frac{16 x^2}{8 y^6}\)[/tex]
- Simplify the coefficients:
[tex]\[
\frac{16}{8} = 2
\][/tex]
- Simplify the [tex]\(x\)[/tex]-terms:
[tex]\[
x^2 \text{ (remains as is)}
\][/tex]
- Simplify the [tex]\(y\)[/tex]-terms:
[tex]\[
\frac{1}{y^6} = y^{-6}
\][/tex]
So, the second fraction simplifies to:
[tex]\[
2 \cdot x^2 \cdot y^{-6} = \frac{2 x^2}{y^6}
\][/tex]
Step 3: Multiply the simplified fractions together:
[tex]\[
\frac{y^4}{8 x^2} \cdot \frac{2 x^2}{y^6}
\][/tex]
Now we simplify this product as follows:
- Simplify the coefficients:
[tex]\[
\frac{1}{8} \cdot 2 = \frac{2}{8} = \frac{1}{4}
\][/tex]
- Simplify the [tex]\(x\)[/tex]-terms:
[tex]\[
x^{-2} \cdot x^2 = x^{-2+2} = x^0 = 1
\][/tex]
- Simplify the [tex]\(y\)[/tex]-terms:
[tex]\[
y^4 \cdot y^{-6} = y^{4-6} = y^{-2} = \frac{1}{y^2}
\][/tex]
Putting it all together:
[tex]\[
\frac{1}{4} \cdot 1 \cdot \frac{1}{y^2} = \frac{1}{4 y^2}
\][/tex]
Thus, the simplified expression is:
[tex]\[
\boxed{\frac{1}{4 y^2}}
\][/tex]
