Without doing any actual division, find which of the rational numbers have terminating decimal representation :
`(23)/(125)`

Answer

To determine whether the rational number \( \frac{23}{125} \) has a terminating decimal representation, we can follow these steps:

### Step 1: Identify the denominator
The given rational number is \( \frac{23}{125} \). Here, the denominator is \( 125 \).

### Step 2: Factor the denominator
Next, we need to factor the denominator \( 125 \) into its prime factors.

\[ 125 = 5 \times 5 \times 5 = 5^3 \]

### Step 3: Check the prime factors
Now, we check the prime factors of \( 125 \). The only prime factor we have is \( 5 \).

### Step 4: Determine if the factors are of the form \( 2^m \times 5^n \)
For a rational number to have a terminating decimal representation, the prime factorization of its denominator (after simplification) must consist only of the primes \( 2 \) and \( 5 \).

In this case, since \( 125 = 5^3 \) and there are no factors of \( 2 \) present, we can express it as:

\[ 125 = 5^3 \times 2^0 \]

### Step 5: Conclusion
Since the denominator \( 125 \) consists only of the prime factor \( 5 \) (and \( 2^0 \)), we conclude that the decimal representation of \( \frac{23}{125} \) is terminating.

### Final Answer
The rational number \( \frac{23}{125} \) has a terminating decimal representation.

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