Why is 64 a perfect square and 60 isnt
The Correct Answer and Explanation is :
A perfect square is defined as a number that can be expressed as the square of an integer. In other words, if a number ( n ) can be written in the form ( n = k^2 ), where ( k ) is an integer, then ( n ) is a perfect square.
Perfect Square of 64
To understand why 64 is a perfect square, we can break it down as follows:
- The number 64 can be factored into prime factors:
[
64 = 2^6
] - Since 64 can be represented as ( 8 \times 8 ) or ( 8^2 ) (where ( 8 ) is an integer), it confirms that 64 is a perfect square.
[
8^2 = 64
]
Non-Perfect Square of 60
In contrast, let’s analyze 60:
- The prime factorization of 60 is:
[
60 = 2^2 \times 3^1 \times 5^1
] - For a number to be a perfect square, all the exponents in its prime factorization must be even. In the case of 60, the prime factorization shows that the exponent of ( 3 ) is ( 1 ) (which is odd) and the exponent of ( 5 ) is also ( 1 ) (also odd).
- Since there are odd exponents in the factorization, 60 cannot be expressed as the square of an integer.
Conclusion
In summary, 64 qualifies as a perfect square because it can be expressed as ( 8^2 ), where 8 is an integer. Meanwhile, 60 lacks this property due to the presence of odd exponents in its prime factorization, preventing it from being expressed as the square of any integer. The distinction between perfect squares and non-perfect squares is rooted in the properties of their prime factorization, specifically the parity (odd or even nature) of the exponents involved.