To simplify the expression
[tex]\[ \frac{4 a^2 b^3 \times 5 b c^3}{10 a b^2 c}, \][/tex]
follow these steps:
1. Multiply the numerators:
[tex]\[
4 a^2 b^3 \times 5 b c^3 = (4 \times 5) \times a^2 \times b^3 \times b \times c^3 = 20 a^2 b^4 c^3
\][/tex]
So, the numerator simplifies to [tex]\( 20 a^2 b^4 c^3 \)[/tex].
2. Combine the numerator with the denominator:
[tex]\[
\frac{20 a^2 b^4 c^3}{10 a b^2 c}
\][/tex]
3. Simplify the coefficients:
[tex]\[
\frac{20}{10} = 2
\][/tex]
4. Simplify the variable [tex]\(a\)[/tex]:
[tex]\[
\frac{a^2}{a} = a^{2-1} = a^1 = a
\][/tex]
5. Simplify the variable [tex]\(b\)[/tex]:
[tex]\[
\frac{b^4}{b^2} = b^{4-2} = b^2
\][/tex]
6. Simplify the variable [tex]\(c\)[/tex]:
[tex]\[
\frac{c^3}{c} = c^{3-1} = c^2
\][/tex]
Putting this all together, the simplified form is:
[tex]\[
2 a b^2 c^2
\][/tex]
Thus, the simplified expression is:
[tex]\[
\boxed{2 a b^2 c^2}
\][/tex]
