Which numbers are perfect squares? Check all that apply.
In mathematics, a perfect square is a number that can be expressed as the square of an integer. This means that the number is the product of a number multiplied by itself. For example, 4 is a perfect square because it can be written as 2 x 2. Similarly, 9 is a perfect square because it is 3 x 3. Let’s explore some of the properties and examples of perfect squares.
Firstly, it is important to note that all perfect squares are non-negative integers. This is because the square of a negative number is always positive. For instance, (-2) x (-2) equals 4, which is a perfect square. However, negative numbers themselves are not considered perfect squares.
One of the key characteristics of perfect squares is that they have an odd number of divisors. For example, the divisors of 4 are 1, 2, and 4. Since there are three divisors, which is an odd number, 4 is a perfect square. In contrast, the divisors of 6 are 1, 2, 3, and 6, which is an even number of divisors, so 6 is not a perfect square.
To determine whether a number is a perfect square, you can use the following methods:
1. Prime factorization: Prime factorize the number and check if all the exponents of the prime factors are even. If they are, then the number is a perfect square. For example, the prime factorization of 36 is 2^2 x 3^2, and since all the exponents are even, 36 is a perfect square.
2. Long division: Divide the number by 1, 2, 3, and so on until the quotient is equal to the divisor. If the quotient is an integer, then the number is a perfect square. For instance, to check if 49 is a perfect square, divide it by 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10. The quotient is 7 when dividing by 7, which is an integer. Therefore, 49 is a perfect square.
3. Square root: Calculate the square root of the number. If the square root is an integer, then the number is a perfect square. For example, the square root of 16 is 4, which is an integer. Hence, 16 is a perfect square.
Now, let’s identify some perfect squares from the given list:
– 4: As mentioned earlier, 4 is a perfect square because it can be written as 2 x 2.
– 9: 9 is a perfect square because it is 3 x 3.
– 16: The square root of 16 is 4, which is an integer. Therefore, 16 is a perfect square.
– 25: 25 is a perfect square because it is 5 x 5.
– 36: The prime factorization of 36 is 2^2 x 3^2, and since all the exponents are even, 36 is a perfect square.
In conclusion, perfect squares are non-negative integers that can be expressed as the square of an integer. To determine whether a number is a perfect square, you can use methods such as prime factorization, long division, or calculating the square root. The given list includes several perfect squares, such as 4, 9, 16, 25, and 36.
