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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]
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]
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