Give reason to show that none of the numbers `640,81000 and 360000` is a perfect square.

Answer

To determine whether the numbers 640, 81,000, and 360,000 are perfect squares, we can analyze the prime factorization of each number and check the conditions for a number to be a perfect square.

### Step-by-Step Solution:

1. **Check the number 640:**
- **Prime Factorization:**
- Divide 640 by 2:
- 640 ÷ 2 = 320
- 320 ÷ 2 = 160
- 160 ÷ 2 = 80
- 80 ÷ 2 = 40
- 40 ÷ 2 = 20
- 20 ÷ 2 = 10
- 10 ÷ 2 = 5
- So, the prime factorization of 640 is \( 2^7 \times 5^1 \).
- **Check for Perfect Square:**
- For a number to be a perfect square, all the exponents in its prime factorization must be even.
- Here, the exponent of 2 is 7 (odd) and the exponent of 5 is 1 (odd).
- Since not all exponents are even, **640 is not a perfect square.**

2. **Check the number 81,000:**
- **Prime Factorization:**
- Divide 81,000 by 10 repeatedly:
- 81,000 ÷ 10 = 8,100
- 8,100 ÷ 10 = 810
- 810 ÷ 10 = 81
- Now factor 81:
- 81 = \( 3^4 \) (since \( 81 = 3 \times 3 \times 3 \times 3 \))
- So, the prime factorization of 81,000 is \( 3^4 \times 10^3 = 3^4 \times (2^1 \times 5^1)^3 = 3^4 \times 2^3 \times 5^3 \).
- **Check for Perfect Square:**
- The exponents are:
- For 3: 4 (even)
- For 2: 3 (odd)
- For 5: 3 (odd)
- Since not all exponents are even, **81,000 is not a perfect square.**

3. **Check the number 360,000:**
- **Prime Factorization:**
- Divide 360,000 by 10 repeatedly:
- 360,000 ÷ 10 = 36,000
- 36,000 ÷ 10 = 3,600
- 3,600 ÷ 10 = 360
- 360 ÷ 10 = 36
- Now factor 36:
- 36 = \( 6^2 = (2 \times 3)^2 = 2^2 \times 3^2 \)
- So, the prime factorization of 360,000 is \( 36 \times 10^4 = (2^2 \times 3^2) \times (2^4 \times 5^4) = 2^6 \times 3^2 \times 5^4 \).
- **Check for Perfect Square:**
- The exponents are:
- For 2: 6 (even)
- For 3: 2 (even)
- For 5: 4 (even)
- Since all exponents are even, **360,000 is a perfect square.**

### Conclusion:
- **640 is not a perfect square.**
- **81,000 is not a perfect square.**
- **360,000 is a perfect square.**