Is 144 a perfect square or not
The Correct Answer and Explanation is:
Yes, 144 is a perfect square. A perfect square is an integer that can be expressed as the product of an integer multiplied by itself. In this case, the number 144 can be expressed as (12 \times 12) or (12^2).
To understand why 144 is a perfect square, we can start with the definition of a perfect square. A perfect square arises when a whole number is squared (multiplied by itself). The general formula for finding a perfect square is:
[
n^2 = n \times n
]
where (n) is a non-negative integer. For instance, if we take (n = 12):
[
12^2 = 12 \times 12 = 144
]
This means that 12 is the square root of 144. When we take the square root of a perfect square, we should obtain a whole number. In this case, the square root of 144 is 12, which confirms that 144 is indeed a perfect square.
We can also verify that 144 is a perfect square by examining its prime factorization. The prime factorization of 144 can be calculated as follows:
- Divide by 2: (144 \div 2 = 72)
- Divide by 2 again: (72 \div 2 = 36)
- Divide by 2 once more: (36 \div 2 = 18)
- Divide by 2 again: (18 \div 2 = 9)
- Divide by 3: (9 \div 3 = 3)
- Divide by 3 again: (3 \div 3 = 1)
The prime factorization of 144 is (2^4 \times 3^2). Since all prime factors are raised to an even power (4 for 2 and 2 for 3), this further confirms that 144 is a perfect square. Thus, the final conclusion is that 144 is indeed a perfect square, represented as (12^2).