The decimal representation of `(129)/(2 ^(2).5 ^(3)) ` will be :

Options

A. terminating
B. Non-terminating
C. Non-terminating and repeating
D. Non-terminating and non-repreating

Answer

To solve the problem of determining the decimal representation of \( \frac{129}{2^2 \cdot 5^3} \), we can follow these steps:

### Step 1: Calculate the denominator
First, we need to calculate the value of the denominator \( 2^2 \cdot 5^3 \).

\[ 2^2 = 4 \]
\[ 5^3 = 125 \]
Now, multiply these two results together:
\[ 2^2 \cdot 5^3 = 4 \cdot 125 = 500 \]

### Step 2: Rewrite the fraction
Now we can rewrite the fraction with the calculated denominator:
\[ \frac{129}{2^2 \cdot 5^3} = \frac{129}{500} \]

### Step 3: Determine the type of decimal representation
Next, we need to determine if this fraction results in a terminating or non-terminating decimal. A fraction has a terminating decimal if its denominator (when in simplest form) consists only of the prime factors 2 and 5.

In this case, the denominator \( 500 \) can be expressed as:
\[ 500 = 2^2 \cdot 5^3 \]
Since the denominator consists only of the prime factors 2 and 5, we conclude that the decimal representation of \( \frac{129}{500} \) is terminating.

### Step 4: Final answer
Thus, the decimal representation of \( \frac{129}{2^2 \cdot 5^3} \) is a terminating decimal.

### Conclusion
The answer is option 1: terminating.

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