Let's simplify the expression step by step:
[tex]\[
\frac{p^2 q^2 r^2}{a^2 b^2} \times \frac{3 a^3 b^3}{4 p^2 q^2 r^3}
\][/tex]
1. Combine the fractions by multiplying the numerators together and the denominators together:
[tex]\[
\frac{(p^2 q^2 r^2) \cdot (3 a^3 b^3)}{(a^2 b^2) \cdot (4 p^2 q^2 r^3)}
\][/tex]
2. Distribute the multiplication in the numerator and the denominator:
Numerator:
[tex]\[
(p^2 q^2 r^2) \cdot (3 a^3 b^3) = 3 p^2 q^2 r^2 a^3 b^3
\][/tex]
Denominator:
[tex]\[
(a^2 b^2) \cdot (4 p^2 q^2 r^3) = 4 a^2 b^2 p^2 q^2 r^3
\][/tex]
Now the expression is:
[tex]\[
\frac{3 p^2 q^2 r^2 a^3 b^3}{4 a^2 b^2 p^2 q^2 r^3}
\][/tex]
3. Cancel out the common terms in the numerator and the denominator:
- [tex]\(p^2\)[/tex] in the numerator cancels with [tex]\(p^2\)[/tex] in the denominator.
- [tex]\(q^2\)[/tex] in the numerator cancels with [tex]\(q^2\)[/tex] in the denominator.
- [tex]\(r^2\)[/tex] in the numerator partially cancels with [tex]\(r^3\)[/tex] in the denominator, leaving [tex]\(r\)[/tex] in the denominator.
- [tex]\(a^2\)[/tex] in the denominator cancels with [tex]\(a^2\)[/tex] (part of [tex]\(a^3\)[/tex]) in the numerator, leaving [tex]\(a\)[/tex] in the numerator.
- [tex]\(b^2\)[/tex] in the denominator cancels with [tex]\(b^2\)[/tex] (part of [tex]\(b^3\)[/tex]) in the numerator, leaving [tex]\(b\)[/tex] in the numerator.
So, the expression reduces to:
[tex]\[
\frac{3 a b}{4 r}
\][/tex]
Therefore, the simplified expression is:
[tex]\[
\boxed{\frac{3 a b}{4 r}}
\][/tex]
