Answer:
To determine the number by which 3888 should be divided to obtain a perfect square, we need to factorize 3888 and then ensure the resulting quotient is a perfect square. Let's go through the steps:
### Step 1: Prime Factorization of 3888
First, we find the prime factors of 3888.
\[ 3888 \div 2 = 1944 \]
\[ 1944 \div 2 = 972 \]
\[ 972 \div 2 = 486 \]
\[ 486 \div 2 = 243 \]
\[ 243 \div 3 = 81 \]
\[ 81 \div 3 = 27 \]
\[ 27 \div 3 = 9 \]
\[ 9 \div 3 = 3 \]
\[ 3 \div 3 = 1 \]
So, the prime factorization of 3888 is:
\[ 3888 = 2^5 \times 3^5 \]
### Step 2: Ensure Perfect Square
For a number to be a perfect square, all the exponents in its prime factorization must be even. Here, we have:
\[ 3888 = 2^5 \times 3^5 \]
We need to divide 3888 by a number that makes all the exponents even.
- For \(2^5\), we need to remove one \(2\) (since \(5-1=4\) is even).
- For \(3^5\), we need to remove one \(3\) (since \(5-1=4\) is even).
So, the number we need to divide by is:
\[ 2 \times 3 = 6 \]
### Step 3: Divide 3888 by 6
\[ \frac{3888}{6} = 648 \]
Now, we have:
\[ 648 = 2^4 \times 3^4 \]
### Step 4: Find the Square Root of 648
Since \(648\) is now a perfect square:
\[ \sqrt{648} = \sqrt{2^4 \times 3^4} \]
\[ \sqrt{648} = \sqrt{(2^2)^2 \times (3^2)^2} \]
\[ \sqrt{648} = 2^2 \times 3^2 \]
\[ \sqrt{648} = 4 \times 9 \]
\[ \sqrt{648} = 36 \]
### Conclusion
The number by which 3888 should be divided to get a perfect square is \(6\), and the square root of the resulting number \(648\) is \(36\).
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Step-by-step explanation:
To determine by what number \( 3888 \) should be divided to get a perfect square, and to find the square root of the resulting number, we can follow these steps:
1. **Prime Factorization of 3888**:
Find the prime factorization of 3888.
\[
3888 \div 2 = 1944 \\
1944 \div 2 = 972 \\
972 \div 2 = 486 \\
486 \div 2 = 243 \\
243 \div 3 = 81 \\
81 \div 3 = 27 \\
27 \div 3 = 9 \\
9 \div 3 = 3 \\
3 \div 3 = 1
\]
Therefore, the prime factorization of \( 3888 \) is:
\[
3888 = 2^5 \times 3^5
\]
2. **Form a Perfect Square**:
To form a perfect square, each exponent in the prime factorization should be even. Currently, both exponents (5 for 2 and 5 for 3) are odd.
- To make \( 2^5 \) a perfect square, divide by \( 2 \) (since \( 2^5 \div 2 = 2^4 \)).
- To make \( 3^5 \) a perfect square, divide by \( 3 \) (since \( 3^5 \div 3 = 3^4 \)).
Therefore, \( 3888 \) should be divided by:
\[
2 \times 3 = 6
\]
3. **Resulting Perfect Square**:
Divide \( 3888 \) by \( 6 \) to get the resulting perfect square.
\[
\frac{3888}{6} = 648
\]
Let's confirm the prime factorization of \( 648 \):
\[
648 = \frac{3888}{6} = 2^4 \times 3^4
\]
Indeed, \( 648 \) is a perfect square since both exponents (4) are even.
4. **Find the Square Root**:
Calculate the square root of \( 648 \).
\[
\sqrt{648} = \sqrt{2^4 \times 3^4} = 2^2 \times 3^2 = 4 \times 9 = 36
\]
Thus, \( 3888 \) should be divided by \( 6 \) to obtain a perfect square, and the square root of the resulting number \( 648 \) is \( 36 \).
