To solve this problem, we need to determine by what number 51200 should be multiplied to get a perfect square and then find the square root of the resulting perfect square.
### Step 1: Prime Factorization
First, let's find the prime factorization of 51200.
51200 can be expressed in its prime factors as:
[tex]\[ 51200 = 2^9 \times 5^2 \][/tex]
Here, the exponent of 2 is 9 (an odd number) and the exponent of 5 is 2 (an even number).
### Step 2: Identifying Multiplicative Factor
To convert 51200 into a perfect square, all the exponents in its prime factorization need to be even.
The exponent of 2 is 9, which is odd. Therefore, we need to multiply by 2 to make the exponent even:
Multiplying the prime factorization by 2, we get:
[tex]\[ 51200 \times 2 = 2^{10} \times 5^2 \][/tex]
Now both exponents (10 for 2 and 2 for 5) are even, making this a perfect square.
### Step 3: Multiplicative Factor
The smallest number to multiply 51200 by to make it a perfect square is 2.
### Step 4: Computing the Perfect Square
Multiplying 51200 by 2, the number becomes:
[tex]\[ 51200 \times 2 = 102400 \][/tex]
### Step 5: Finding the Square Root
Now, let's find the square root of 102400.
Since:
[tex]\[ 102400 = 2^{10} \times 5^2 \][/tex]
Taking the square root of both sides:
[tex]\[ \sqrt{102400} = \sqrt{2^{10} \times 5^2} = 2^{5} \times 5 = 32 \times 5 = 160 \][/tex]
Therefore, after multiplying 51200 by 2, the resulting perfect square is 102400, and the square root of 102400 is 160.
### Final Answer
The number by which 51200 should be multiplied to get a perfect square is 2.
The resulting perfect square is 102400.
The square root of the number obtained is 160.
