Answer :
To solve this problem, we will break down each number into its prime factors and then determine the least number by which to divide each to make it a perfect square. Additionally, we will find the square root of the resulting perfect square.
### (a) 1250
1. Prime factorization of 1250:
\( 1250 = 2 \times 5^4 \)
Here, \( 5^4 \) is already a perfect square (since 4 is even), but 2 (which has a power of 1) is not.
2. Least number to divide:
The factor with an odd power is 2.
Therefore, the least number by which 1250 should be divided to make it a perfect square is 2.
3. Resulting perfect square:
\( 1250 \div 2 = 625 \)
4. Square root of 625:
\( \sqrt{625} = 25 \)
### (b) 2048
1. Prime factorization of 2048:
\( 2048 = 2^{11} \)
2048 is already in the form of \( 2^{11} \), which is not a perfect square because 11 is odd.
2. Least number to divide:
The factor with an odd power is 2.
Therefore, the least number by which 2048 should be divided to make it a perfect square is 2.
3. Resulting perfect square:
\( 2048 \div 2 = 1024 \)
4. Square root of 1024:
\( \sqrt{1024} = 32 \)
### (c) 2187
1. Prime factorization of 2187:
\( 2187 = 3^7 \)
Here, \( 3^7 \) does not form a perfect square because 7 is odd.
2. Least number to divide:
The factor with an odd power is 3.
Therefore, the least number by which 2187 should be divided to make it a perfect square is 3.
3. Resulting perfect square:
\( 2187 \div 3 = 729 \)
4. Square root of 729:
\( \sqrt{729} = 27 \)
### (d) 7200
1. Prime factorization of 7200:
\( 7200 = 2^4 \times 3^2 \times 5^2 \times 2^2 \)
Here, if we consider the correct analysis, it simplifies to \( 7200 = 2^4 \times 3^1 \times 5^2 \times 2^2 \).
Combining the factors of 2: \( 2^6 \times 3^1 \times 5^2 \).
2. Least number to divide:
The factor with an odd power is 3.
Therefore, the least number by which 7200 should be divided to make it a perfect square is 2.
3. Resulting perfect square:
\( 7200 \div 2 = 3600 \)
4. Square root of 3600:
\( \sqrt{3600} = 60 \)
### Summary:
- (a) \( 1250 \div 2 = 625 \), \( \sqrt{625} = 25 \)
- (b) \( 2048 \div 2 = 1024 \), \( \sqrt{1024} = 32 \)
- (c) \( 2187 \div 3 = 729 \), \( \sqrt{729} = 27 \)
- (d) [tex]\( 7200 \div 2 = 3600 \)[/tex], [tex]\( \sqrt{3600} = 60 \)[/tex]
### (a) 1250
1. Prime factorization of 1250:
\( 1250 = 2 \times 5^4 \)
Here, \( 5^4 \) is already a perfect square (since 4 is even), but 2 (which has a power of 1) is not.
2. Least number to divide:
The factor with an odd power is 2.
Therefore, the least number by which 1250 should be divided to make it a perfect square is 2.
3. Resulting perfect square:
\( 1250 \div 2 = 625 \)
4. Square root of 625:
\( \sqrt{625} = 25 \)
### (b) 2048
1. Prime factorization of 2048:
\( 2048 = 2^{11} \)
2048 is already in the form of \( 2^{11} \), which is not a perfect square because 11 is odd.
2. Least number to divide:
The factor with an odd power is 2.
Therefore, the least number by which 2048 should be divided to make it a perfect square is 2.
3. Resulting perfect square:
\( 2048 \div 2 = 1024 \)
4. Square root of 1024:
\( \sqrt{1024} = 32 \)
### (c) 2187
1. Prime factorization of 2187:
\( 2187 = 3^7 \)
Here, \( 3^7 \) does not form a perfect square because 7 is odd.
2. Least number to divide:
The factor with an odd power is 3.
Therefore, the least number by which 2187 should be divided to make it a perfect square is 3.
3. Resulting perfect square:
\( 2187 \div 3 = 729 \)
4. Square root of 729:
\( \sqrt{729} = 27 \)
### (d) 7200
1. Prime factorization of 7200:
\( 7200 = 2^4 \times 3^2 \times 5^2 \times 2^2 \)
Here, if we consider the correct analysis, it simplifies to \( 7200 = 2^4 \times 3^1 \times 5^2 \times 2^2 \).
Combining the factors of 2: \( 2^6 \times 3^1 \times 5^2 \).
2. Least number to divide:
The factor with an odd power is 3.
Therefore, the least number by which 7200 should be divided to make it a perfect square is 2.
3. Resulting perfect square:
\( 7200 \div 2 = 3600 \)
4. Square root of 3600:
\( \sqrt{3600} = 60 \)
### Summary:
- (a) \( 1250 \div 2 = 625 \), \( \sqrt{625} = 25 \)
- (b) \( 2048 \div 2 = 1024 \), \( \sqrt{1024} = 32 \)
- (c) \( 2187 \div 3 = 729 \), \( \sqrt{729} = 27 \)
- (d) [tex]\( 7200 \div 2 = 3600 \)[/tex], [tex]\( \sqrt{3600} = 60 \)[/tex]
