To solve the problem of finding the smallest number by which 9408 must be divided to make it a perfect square, we need to follow a systematic approach. Let's go through the steps in detail.
### Step 1: Prime Factorization
Firstly, we need to perform the prime factorization of 9408. The prime factorization of 9408 is:
[tex]\[ 9408 = 2^6 \times 3^1 \times 7^2 \][/tex]
### Step 2: Identify Odd Exponents
For a number to be a perfect square, all the exponents in its prime factorization must be even. Here we have an exponent of 1 for the prime number 3, which is odd. This indicates that 3 is a factor we need to deal with, as its exponent prevents 9408 from being a perfect square.
### Step 3: Find the Smallest Divisor
To make 9408 a perfect square, we must divide it by the smallest number that will ensure all exponents are even. This number should include the prime factor with the odd exponent. Therefore, the smallest number to divide by is 3.
### Step 4: Calculate the Resulting Perfect Square
Dividing 9408 by 3, we get:
[tex]\[ \text{Resulting Perfect Square} = \frac{9408}{3} = 3136 \][/tex]
### Step 5: Verify if the Result is a Perfect Square
Next, we need to check if 3136 is indeed a perfect square. We can do this by either performing the prime factorization of 3136 again or simply calculating its square root. The square root of 3136 is:
[tex]\[ \sqrt{3136} = 56 \][/tex]
### Conclusion
Thus, the smallest number by which 9408 must be divided to obtain a perfect square is 3, and the square root of the resulting perfect square (3136) is 56.
### Summary
- Smallest number to divide by: 3
- Square root of the perfect square obtained: 56
