To find the value of the expression [tex]\(\frac{m n^2 + m^2 n^2}{m n + n}\)[/tex] given that [tex]\(M = 3\)[/tex] and [tex]\(\pi = 2\)[/tex], we start by setting [tex]\(m = M = 3\)[/tex] and [tex]\(n = \pi = 2\)[/tex].
Now we plug these values into the expression, which is
[tex]\[
\frac{m n^2 + m^2 n^2}{m n + n}.
\][/tex]
First, let's determine the numerator:
[tex]\[
\text{Numerator} = m n^2 + m^2 n^2.
\][/tex]
Substituting [tex]\(m = 3\)[/tex] and [tex]\(n = 2\)[/tex]:
[tex]\[
m n^2 = 3 \cdot (2)^2 = 3 \cdot 4 = 12,
\][/tex]
[tex]\[
m^2 n^2 = (3)^2 \cdot (2)^2 = 9 \cdot 4 = 36.
\][/tex]
Adding these together:
[tex]\[
m n^2 + m^2 n^2 = 12 + 36 = 48.
\][/tex]
Next, let's determine the denominator:
[tex]\[
\text{Denominator} = m n + n.
\][/tex]
Substituting [tex]\(m = 3\)[/tex] and [tex]\(n = 2\)[/tex]:
[tex]\[
m n = 3 \cdot 2 = 6,
\][/tex]
[tex]\[
n = 2.
\][/tex]
Adding these together:
[tex]\[
m n + n = 6 + 2 = 8.
\][/tex]
Now, we divide the numerator by the denominator:
[tex]\[
\frac{m n^2 + m^2 n^2}{m n + n} = \frac{48}{8}.
\][/tex]
Simplifying this fraction:
[tex]\[
\frac{48}{8} = 6.
\][/tex]
Therefore, the value of the expression is:
[tex]\[
6.0.
\][/tex]
