Simplify : `((21)/(16) xx (12)/(9)) div ((-3)/(8) xx ((-12)/(9)))`

Answer

To simplify the expression \(\frac{21}{16} \times \frac{12}{9} \div \left( \frac{-3}{8} \times \frac{-12}{9} \right)\), we can follow these steps:

### Step 1: Simplify the first part \(\frac{21}{16} \times \frac{12}{9}\)

1. **Factor the fractions**:
- \(\frac{21}{16} = \frac{21}{4 \times 4}\)
- \(\frac{12}{9} = \frac{4 \times 3}{3 \times 3}\)

2. **Cancel common factors**:
- The \(3\) in the numerator of \(\frac{12}{9}\) cancels with one \(3\) in the denominator of \(\frac{12}{9}\).
- The \(4\) in the numerator of \(\frac{12}{9}\) cancels with \(16\) in the denominator of \(\frac{21}{16}\).

3. **Calculate the simplified fraction**:
- After canceling, we have:
\[ \frac{7}{4} \]

### Step 2: Simplify the second part \(\frac{-3}{8} \times \frac{-12}{9}\)

1. **Factor the fractions**:
- \(\frac{-3}{8} = \frac{-3}{2 \times 4}\)
- \(\frac{-12}{9} = \frac{-4 \times 3}{3 \times 3}\)

2. **Cancel common factors**:
- The \(3\) in the numerator of \(\frac{-12}{9}\) cancels with one \(3\) in the denominator of \(\frac{-12}{9}\).
- The \(4\) in the numerator of \(\frac{-12}{9}\) cancels with \(8\) in the denominator of \(\frac{-3}{8}\).

3. **Calculate the simplified fraction**:
- After canceling, we have:
\[ \frac{1}{2} \]

### Step 3: Rewrite the expression

Now, we rewrite the original expression with the simplified fractions:
\[ \frac{7}{4} \div \frac{1}{2} \]

### Step 4: Change division to multiplication

To divide by a fraction, we multiply by its reciprocal:
\[ \frac{7}{4} \times \frac{2}{1} \]

### Step 5: Multiply the fractions

Now, we multiply the numerators and denominators:
\[ \frac{7 \times 2}{4 \times 1} = \frac{14}{4} \]

### Step 6: Simplify the final fraction

Now, we simplify \(\frac{14}{4}\):
- Divide both numerator and denominator by \(2\):
\[ \frac{14 \div 2}{4 \div 2} = \frac{7}{2} \]

### Final Answer
The simplified expression is:
\[ \frac{7}{2} \]

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