Solve : simplify `(4)/(3)m^(2)-(3)/(4)n^(2)+2mn-((16)/(9)m^(2)+(9)/(16)n^(2)+2mn)`.

Answer

To simplify the expression \( \frac{4}{3}m^2 - \frac{3}{4}n^2 + 2mn - \left( \frac{16}{9}m^2 + \frac{9}{16}n^2 + 2mn \right) \), we will follow these steps:

### Step 1: Distribute the negative sign
We start by distributing the negative sign across the terms inside the parentheses:

\[ \frac{4}{3}m^2 - \frac{3}{4}n^2 + 2mn - \frac{16}{9}m^2 - \frac{9}{16}n^2 - 2mn \]

### Step 2: Combine like terms
Now, we will combine the like terms. The like terms here are the \( m^2 \) terms, the \( n^2 \) terms, and the \( mn \) terms.

1. **Combine \( m^2 \) terms**:
\[ \frac{4}{3}m^2 - \frac{16}{9}m^2 \]

To combine these, we need a common denominator. The least common multiple of 3 and 9 is 9. We convert \( \frac{4}{3} \) to have a denominator of 9:

\[ \frac{4}{3} = \frac{4 \times 3}{3 \times 3} = \frac{12}{9} \]

Now we can combine:
\[ \frac{12}{9}m^2 - \frac{16}{9}m^2 = \frac{12 - 16}{9}m^2 = -\frac{4}{9}m^2 \]

2. **Combine \( n^2 \) terms**:
\[ -\frac{3}{4}n^2 - \frac{9}{16}n^2 \]

The least common multiple of 4 and 16 is 16. We convert \( -\frac{3}{4} \) to have a denominator of 16:
\[ -\frac{3}{4} = -\frac{3 \times 4}{4 \times 4} = -\frac{12}{16} \]

Now we can combine:
\[ -\frac{12}{16}n^2 - \frac{9}{16}n^2 = -\frac{12 + 9}{16}n^2 = -\frac{21}{16}n^2 \]

3. **Combine \( mn \) terms**:
\[ 2mn - 2mn = 0 \]

### Step 3: Write the final simplified expression
Putting it all together, we have:
\[ -\frac{4}{9}m^2 - \frac{21}{16}n^2 \]

### Final Answer:
The simplified expression is:
\[ -\frac{4}{9}m^2 - \frac{21}{16}n^2 \]