Algebraic Expression Analysis

The problem asks us to find the value of a complex expression given two initial equations involving sums of natural numbers. We need to simplify the given equations first to find relationships between the variables \({\rm{a}}\), \({\rm{b}}\), \({\rm{n}}\), and \({\rm{m}}\).

Understanding Given Equations

Let's break down the two equations provided:

1. The first equation is \(\underbrace {{\rm{a}} + {\rm{a}} + {\rm{a}} + \ldots + {\rm{a}}}_{{\rm{n\;times}}} = {{\rm{a}}^2}{\rm{b}}\).
2. The second equation is \(\underbrace {{\rm{b}} + {\rm{b}} + {\rm{b}} + \ldots + {\rm{b}}}_{{\rm{m\;times}}} = {\rm{a}}{{\rm{b}}^2}\).

Here, \({\rm{a}}\), \({\rm{b}}\), \({\rm{n}}\) and \({\rm{m}}\) are natural numbers. This means they are positive integers (1, 2, 3, ...).

Simplifying Initial Equations

Equation 1 Simplification

The sum of \({\rm{a}}\) added \({\rm{n}}\) times can be written as \({\rm{n}} \times {\rm{a}}\) or \({\rm{na}}\). So, the first equation becomes:

\[ {\rm{na}} = {{\rm{a}}^2}{\rm{b}} \]

Since \({\rm{a}}\) is a natural number, \({\rm{a}} \neq 0\). We can divide both sides of the equation by \({\rm{a}}\):

\[ {\rm{n}} = {\rm{ab}} \quad \text{(Equation A)} \]

Equation 2 Simplification

Similarly, the sum of \({\rm{b}}\) added \({\rm{m}}\) times can be written as \({\rm{m}} \times {\rm{b}}\) or \({\rm{mb}}\). So, the second equation becomes:

\[ {\rm{mb}} = {\rm{a}}{{\rm{b}}^2} \]

Since \({\rm{b}}\) is a natural number, \({\rm{b}} \neq 0\). We can divide both sides of the equation by \({\rm{b}}\):

\[ {\rm{m}} = {\rm{ab}} \quad \text{(Equation B)} \]

Relating the Variables

From Equation A and Equation B, we observe a significant relationship:

• \({\rm{n}} = {\rm{ab}}\)
• \({\rm{m}} = {\rm{ab}}\)

This implies that \({\rm{n}} = {\rm{m}}\).

Calculating the Final Expression

We need to find the value of the expression \(\left( {\underbrace {{\rm{m}} + {\rm{m}} + {\rm{m}} + \ldots + {\rm{m}}}_{{\rm{n\;times}}}} \right)\left( {\underbrace {{\rm{n}} + {\rm{n}} + {\rm{n}} + \ldots + {\rm{n}}}_{{\rm{m\;times}}}} \right)\).

Let's simplify each part of the product:

• The first part, \(\underbrace {{\rm{m}} + {\rm{m}} + {\rm{m}} + \ldots + {\rm{m}}}_{{\rm{n\;times}}}\), is \({\rm{m}}\) added \({\rm{n}}\) times, which equals \({\rm{n}} \times {\rm{m}}\) or \({\rm{nm}}\).
• The second part, \(\underbrace {{\rm{n}} + {\rm{n}} + {\rm{n}} + \ldots + {\rm{n}}}_{{\rm{m\;times}}}\), is \({\rm{n}}\) added \({\rm{m}}\) times, which equals \({\rm{m}} \times {\rm{n}}\) or \({\rm{mn}}\).

So, the entire expression becomes:

\[ \left( {{\rm{nm}}} \right)\left( {{\rm{mn}}} \right) = \left( {{\rm{nm}}} \right)^2 \]

Now, substitute the values of \({\rm{n}}\) and \({\rm{m}}\) we found from Equation A and Equation B:

• Substitute \({\rm{n}} = {\rm{ab}}\)
• Substitute \({\rm{m}} = {\rm{ab}}\)

\[ \left( {{\rm{nm}}} \right)^2 = \left( \left( {\rm{ab}} \right) \left( {\rm{ab}} \right) \right)^2 \]

First, multiply the terms inside the parentheses:

\[ \left( {\left( {\rm{ab}} \right)\left( {\rm{ab}} \right)} \right) = \left( {{\rm{a}} \cdot {\rm{a}} \cdot {\rm{b}} \cdot {\rm{b}}} \right) = \left( {{{\rm{a}}^2}{{\rm{b}}^2}} \right) \]

Now, square this result:

\[ \left( {{{\rm{a}}^2}{{\rm{b}}^2}} \right)^2 = {{\rm{a}}^{2 \times 2}}{{\rm{b}}^{2 \times 2}} = {{\rm{a}}^4}{{\rm{b}}^4} \]

Therefore, the value of the given expression is \({\rm{a}}^4{{\rm{b}}^4}\).

Conclusion

By simplifying the initial equations, we found that \({\rm{n}} = {\rm{ab}}\) and \({\rm{m}} = {\rm{ab}}\). Substituting these relationships into the target expression, we were able to simplify it down to \({\rm{a}}^4{{\rm{b}}^4}\).