Without doing any actual division, find which of the rational numbers have terminating decimal representation :
`(9)/(14)`

Answer

To determine whether the rational number \( \frac{9}{14} \) has a terminating decimal representation, we can follow these steps:

### Step 1: Factor the Denominator
First, we need to factor the denominator of the rational number. The denominator here is \( 14 \).

\[ 14 = 2 \times 7 \]

### Step 2: Identify the Prime Factors
Next, we identify the prime factors of the denominator. From our factorization, we see that the prime factors of \( 14 \) are \( 2 \) and \( 7 \).

### Step 3: Check the Form of the Denominator
For a rational number to have a terminating decimal representation, its denominator (after simplification) must be in the form of \( 2^m \times 5^n \), where \( m \) and \( n \) are non-negative integers (natural numbers including zero).

In our case, the denominator \( 14 = 2^1 \times 7^1 \). Here, we have a factor of \( 7 \) which is not allowed for a terminating decimal representation.

### Step 4: Conclusion
Since the denominator \( 14 \) contains a prime factor \( 7 \) (which is neither \( 2 \) nor \( 5 \)), we conclude that the rational number \( \frac{9}{14} \) does not have a terminating decimal representation.

### Final Answer
The rational number \( \frac{9}{14} \) does not have a terminating decimal representation.

---