Without actually performing the long division, state whether the rational number will have a terminating decimal expansion or a non - terminating repeating decimal expansion
`(15)/(1600)`

Answer

To determine whether the rational number \( \frac{15}{1600} \) has a terminating decimal expansion or a non-terminating repeating decimal expansion, we can follow these steps:

### Step 1: Simplify the Rational Number
First, we simplify the fraction \( \frac{15}{1600} \) to its lowest terms.

- The greatest common divisor (GCD) of 15 and 1600 is 5.
- Dividing both the numerator and the denominator by 5:
\[ \frac{15 \div 5}{1600 \div 5} = \frac{3}{320} \]

### Step 2: Factor the Denominator
Next, we need to factor the denominator (320) into its prime factors to check if it only contains the prime factors 2 and 5.

- We can find the prime factorization of 320:
\[ 320 = 32 \times 10 = 2^5 \times (2 \times 5) = 2^6 \times 5^1 \]

### Step 3: Check the Prime Factors
Now we check the prime factors of the denominator:

- The prime factorization of 320 is \( 2^6 \times 5^1 \).
- Since the only prime factors in the denominator are 2 and 5, we can conclude that the decimal expansion of the fraction will be terminating.

### Conclusion
Thus, the rational number \( \frac{15}{1600} \) has a terminating decimal expansion.

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