Without actual division, show that each of the following rational numbers is a terminating decimal. Express each in decimal form.
` 23/((2^(3) xx 5^(2))` (ii) ` 24/125` (iii) ` 171/800` (iv) `15/1600`
(v) ` 17/320` (vi)` 19/3125`

Explanation

To determine whether the given rational numbers are terminating decimals and to express them in decimal form, we will follow these steps:

### Step-by-Step Solution:

1. **Identify the Form of the Denominator**:
A rational number in the form \( \frac{a}{b} \) is a terminating decimal if the denominator \( b \) can be expressed as \( 2^m \times 5^n \) for non-negative integers \( m \) and \( n \).

2. **Calculate Each Rational Number**:

**(i)** \( \frac{23}{2^3 \times 5^2} \)

- The denominator is \( 2^3 \times 5^2 \).
- To make the powers of 2 and 5 equal, we multiply the numerator and denominator by \( 5^1 \):
\[ \frac{23 \times 5}{2^3 \times 5^3} = \frac{115}{1000} \]
- Decimal form: \( 0.115 \)

**(ii)** \( \frac{24}{125} \)

- The denominator \( 125 = 5^3 \) can be expressed as \( 2^0 \times 5^3 \).
- To make the powers equal, we multiply the numerator and denominator by \( 2^3 \):
\[ \frac{24 \times 8}{2^3 \times 5^3} = \frac{192}{1000} \]
- Decimal form: \( 0.192 \)

**(iii)** \( \frac{171}{800} \)

- The denominator \( 800 = 2^5 \times 5^2 \).
- To make the powers equal, we multiply the numerator and denominator by \( 5^3 \):
\[ \frac{171 \times 125}{2^5 \times 5^5} = \frac{21375}{100000} \]
- Decimal form: \( 0.21375 \)

**(iv)** \( \frac{15}{1600} \)

- The denominator \( 1600 = 2^6 \times 5^2 \).
- To make the powers equal, we multiply the numerator and denominator by \( 5^4 \):
\[ \frac{15 \times 625}{2^6 \times 5^6} = \frac{9375}{100000} \]
- Decimal form: \( 0.09375 \)

**(v)** \( \frac{17}{320} \)

- The denominator \( 320 = 2^6 \times 5^1 \).
- To make the powers equal, we multiply the numerator and denominator by \( 5^5 \):
\[ \frac{17 \times 3125}{2^6 \times 5^6} = \frac{53125}{1000000} \]
- Decimal form: \( 0.053125 \)

**(vi)** \( \frac{19}{3125} \)

- The denominator \( 3125 = 5^5 \) can be expressed as \( 2^0 \times 5^5 \).
- To make the powers equal, we multiply the numerator and denominator by \( 2^5 \):
\[ \frac{19 \times 32}{2^5 \times 5^5} = \frac{608}{100000} \]
- Decimal form: \( 0.00608 \)

### Final Answers:
1. \( \frac{23}{2^3 \times 5^2} = 0.115 \)
2. \( \frac{24}{125} = 0.192 \)
3. \( \frac{171}{800} = 0.21375 \)
4. \( \frac{15}{1600} = 0.09375 \)
5. \( \frac{17}{320} = 0.053125 \)
6. \( \frac{19}{3125} = 0.00608 \)