Answer :
To solve this problem, we need to determine a specific number by which 3888 should be divided to obtain a perfect square, and then find the square root of that perfect square. Here’s a detailed, step-by-step solution:
1. Prime Factorization of 3888:
- Prime factorization involves expressing 3888 as a product of prime numbers.
- The prime factors of 3888 are: [tex]\( 2^4 \times 3^5 \)[/tex].
2. Identify Factors with Odd Exponents:
- For a number to be a perfect square, all prime factors must have even exponents when expressed in their prime factorized form.
- In [tex]\( 3888 \)[/tex], the factor [tex]\( 2 \)[/tex] has an exponent of 4 (even), and the factor [tex]\( 3 \)[/tex] has an exponent of 5 (odd).
3. Determine the Number to Divide:
- To make the number a perfect square, we need to get rid of any factors with odd exponents.
- Here, [tex]\( 3 \)[/tex] has an odd exponent (5), so we need to divide by 3 to make the remaining product a perfect square.
4. Calculate the Perfect Square:
- We divide 3888 by 3:
[tex]\[ \text{Perfect square} = \frac{3888}{3} = 1296 \][/tex]
5. Find the Square Root:
- The square root of 1296 is calculated as:
[tex]\[ \sqrt{1296} = 36 \][/tex]
So, the number by which 3888 should be divided to get a perfect square is 3. The resulting perfect square is 1296, and the square root of this perfect square is 36.
1. Prime Factorization of 3888:
- Prime factorization involves expressing 3888 as a product of prime numbers.
- The prime factors of 3888 are: [tex]\( 2^4 \times 3^5 \)[/tex].
2. Identify Factors with Odd Exponents:
- For a number to be a perfect square, all prime factors must have even exponents when expressed in their prime factorized form.
- In [tex]\( 3888 \)[/tex], the factor [tex]\( 2 \)[/tex] has an exponent of 4 (even), and the factor [tex]\( 3 \)[/tex] has an exponent of 5 (odd).
3. Determine the Number to Divide:
- To make the number a perfect square, we need to get rid of any factors with odd exponents.
- Here, [tex]\( 3 \)[/tex] has an odd exponent (5), so we need to divide by 3 to make the remaining product a perfect square.
4. Calculate the Perfect Square:
- We divide 3888 by 3:
[tex]\[ \text{Perfect square} = \frac{3888}{3} = 1296 \][/tex]
5. Find the Square Root:
- The square root of 1296 is calculated as:
[tex]\[ \sqrt{1296} = 36 \][/tex]
So, the number by which 3888 should be divided to get a perfect square is 3. The resulting perfect square is 1296, and the square root of this perfect square is 36.
