To determine whether 176400 is a square number, we need to examine its prime factorization.
Given the prime factorization:
[tex]\[ 176400 = 2^4 \times 3^2 \times 5^2 \times 7^2 \][/tex]
### Step-by-Step Analysis:
1. Understanding Prime Factorization:
- The factor [tex]\( 2 \)[/tex] appears with an exponent of 4: [tex]\( 2^4 \)[/tex]
- The factor [tex]\( 3 \)[/tex] appears with an exponent of 2: [tex]\( 3^2 \)[/tex]
- The factor [tex]\( 5 \)[/tex] appears with an exponent of 2: [tex]\( 5^2 \)[/tex]
- The factor [tex]\( 7 \)[/tex] appears with an exponent of 2: [tex]\( 7^2 \)[/tex]
2. Square Numbers and Exponents:
- A number is a perfect square if all the exponents in its prime factorization are even.
3. Checking Each Exponent:
- For [tex]\( 2^4 \)[/tex], the exponent is 4, which is even.
- For [tex]\( 3^2 \)[/tex], the exponent is 2, which is even.
- For [tex]\( 5^2 \)[/tex], the exponent is 2, which is even.
- For [tex]\( 7^2 \)[/tex], the exponent is 2, which is even.
4. Conclusion:
- Since all the exponents (4 for 2, 2 for 3, 2 for 5, and 2 for 7) are even, this confirms that 176400 is indeed a square number.
### Summary:
Based on the prime factorization [tex]\( 176400 = 2^4 \times 3^2 \times 5^2 \times 7^2 \)[/tex], where all exponents are even, we can conclude that 176400 is a perfect square.
