Answer :
Sure! Let's evaluate which of the given numbers are not perfect squares by using the prime factorization method. A perfect square will have all even powers in its prime factorization.
### Step-by-Step Solution
#### (i) 324
1. Prime factorize 324:
- 324 is divisible by 2: \( 324 \div 2 = 162 \).
- 162 is divisible by 2: \( 162 \div 2 = 81 \).
- 81 is divisible by 3: \( 81 \div 3 = 27 \).
- 27 is divisible by 3: \( 27 \div 3 = 9 \).
- 9 is divisible by 3: \( 9 \div 3 = 3 \).
- 3 is divisible by 3: \( 3 \div 3 = 1 \).
So, \( 324 = 2^2 \times 3^4 \).
2. Check the powers of the prime factors:
- The power of 2 is 2.
- The power of 3 is 4.
Both powers are even, hence 324 is a perfect square.
#### (ii) 768
1. Prime factorize 768:
- 768 is divisible by 2: \( 768 \div 2 = 384 \).
- 384 is divisible by 2: \( 384 \div 2 = 192 \).
- 192 is divisible by 2: \( 192 \div 2 = 96 \).
- 96 is divisible by 2: \( 96 \div 2 = 48 \).
- 48 is divisible by 2: \( 48 \div 2 = 24 \).
- 24 is divisible by 2: \( 24 \div 2 = 12 \).
- 12 is divisible by 2: \( 12 \div 2 = 6 \).
- 6 is divisible by 2: \( 6 \div 2 = 3 \).
- 3 is divisible by 3: \( 3 \div 3 = 1 \).
So, \( 768 = 2^8 \times 3^1 \).
2. Check the powers of the prime factors:
- The power of 2 is 8 (even).
- The power of 3 is 1 (odd).
Since one of the powers is odd, 768 is not a perfect square.
#### (iii) 1296
1. Prime factorize 1296:
- 1296 is divisible by 2: \( 1296 \div 2 = 648 \).
- 648 is divisible by 2: \( 648 \div 2 = 324 \).
- 324 is divisible by 2: \( 324 \div 2 = 162 \).
- 162 is divisible by 2: \( 162 \div 2 = 81 \).
- 81 is divisible by 3: \( 81 \div 3 = 27 \).
- 27 is divisible by 3: \( 27 \div 3 = 9 \).
- 9 is divisible by 3: \( 9 \div 3 = 3 \).
- 3 is divisible by 3: \( 3 \div 3 = 1 \).
So, \( 1296 = 2^4 \times 3^4 \).
2. Check the powers of the prime factors:
- The power of 2 is 4.
- The power of 3 is 4.
Both powers are even, hence 1296 is a perfect square.
#### (iv) 8000
1. Prime factorize 8000:
- 8000 is divisible by 2: \( 8000 \div 2 = 4000 \).
- 4000 is divisible by 2: \( 4000 \div 2 = 2000 \).
- 2000 is divisible by 2: \( 2000 \div 2 = 1000 \).
- 1000 is divisible by 2: \( 1000 \div 2 = 500 \).
- 500 is divisible by 2: \( 500 \div 2 = 250 \).
- 250 is divisible by 2: \( 250 \div 2 = 125 \).
- 125 is divisible by 5: \( 125 \div 5 = 25 \).
- 25 is divisible by 5: \( 25 \div 5 = 5 \).
- 5 is divisible by 5: \( 5 \div 5 = 1 \).
So, \( 8000 = 2^6 \times 5^3 \).
2. Check the powers of the prime factors:
- The power of 2 is 6 (even).
- The power of 5 is 3 (odd).
Since one of the powers is odd, 8000 is not a perfect square.
### Conclusion
- 324 and 1296 are perfect squares.
- 768 and 8000 are not perfect squares.
Therefore, the numbers that are not perfect squares are:
- (ii) 768
- (iv) 8000
### Step-by-Step Solution
#### (i) 324
1. Prime factorize 324:
- 324 is divisible by 2: \( 324 \div 2 = 162 \).
- 162 is divisible by 2: \( 162 \div 2 = 81 \).
- 81 is divisible by 3: \( 81 \div 3 = 27 \).
- 27 is divisible by 3: \( 27 \div 3 = 9 \).
- 9 is divisible by 3: \( 9 \div 3 = 3 \).
- 3 is divisible by 3: \( 3 \div 3 = 1 \).
So, \( 324 = 2^2 \times 3^4 \).
2. Check the powers of the prime factors:
- The power of 2 is 2.
- The power of 3 is 4.
Both powers are even, hence 324 is a perfect square.
#### (ii) 768
1. Prime factorize 768:
- 768 is divisible by 2: \( 768 \div 2 = 384 \).
- 384 is divisible by 2: \( 384 \div 2 = 192 \).
- 192 is divisible by 2: \( 192 \div 2 = 96 \).
- 96 is divisible by 2: \( 96 \div 2 = 48 \).
- 48 is divisible by 2: \( 48 \div 2 = 24 \).
- 24 is divisible by 2: \( 24 \div 2 = 12 \).
- 12 is divisible by 2: \( 12 \div 2 = 6 \).
- 6 is divisible by 2: \( 6 \div 2 = 3 \).
- 3 is divisible by 3: \( 3 \div 3 = 1 \).
So, \( 768 = 2^8 \times 3^1 \).
2. Check the powers of the prime factors:
- The power of 2 is 8 (even).
- The power of 3 is 1 (odd).
Since one of the powers is odd, 768 is not a perfect square.
#### (iii) 1296
1. Prime factorize 1296:
- 1296 is divisible by 2: \( 1296 \div 2 = 648 \).
- 648 is divisible by 2: \( 648 \div 2 = 324 \).
- 324 is divisible by 2: \( 324 \div 2 = 162 \).
- 162 is divisible by 2: \( 162 \div 2 = 81 \).
- 81 is divisible by 3: \( 81 \div 3 = 27 \).
- 27 is divisible by 3: \( 27 \div 3 = 9 \).
- 9 is divisible by 3: \( 9 \div 3 = 3 \).
- 3 is divisible by 3: \( 3 \div 3 = 1 \).
So, \( 1296 = 2^4 \times 3^4 \).
2. Check the powers of the prime factors:
- The power of 2 is 4.
- The power of 3 is 4.
Both powers are even, hence 1296 is a perfect square.
#### (iv) 8000
1. Prime factorize 8000:
- 8000 is divisible by 2: \( 8000 \div 2 = 4000 \).
- 4000 is divisible by 2: \( 4000 \div 2 = 2000 \).
- 2000 is divisible by 2: \( 2000 \div 2 = 1000 \).
- 1000 is divisible by 2: \( 1000 \div 2 = 500 \).
- 500 is divisible by 2: \( 500 \div 2 = 250 \).
- 250 is divisible by 2: \( 250 \div 2 = 125 \).
- 125 is divisible by 5: \( 125 \div 5 = 25 \).
- 25 is divisible by 5: \( 25 \div 5 = 5 \).
- 5 is divisible by 5: \( 5 \div 5 = 1 \).
So, \( 8000 = 2^6 \times 5^3 \).
2. Check the powers of the prime factors:
- The power of 2 is 6 (even).
- The power of 5 is 3 (odd).
Since one of the powers is odd, 8000 is not a perfect square.
### Conclusion
- 324 and 1296 are perfect squares.
- 768 and 8000 are not perfect squares.
Therefore, the numbers that are not perfect squares are:
- (ii) 768
- (iv) 8000
