You can write a perfect square A as the square of another integer a: A=a^2.
Is the product always a perfect square? Yes, see solution. Is the quotient always a perfect square? No, see solution.
Practice makes perfect
For a number to be a perfect square , its square root must be an integer. This means that if an integer A is a perfect square it can be written as A = a^2, where a is another integer.
&sqrt(A)= sqrt(a^2)
&sqrt(A)= a
Let's consider the product of two perfect squares.
AB = a^2b^2For simplicity we are considering that A, B, a, and b are positive integers. However, the result we will obtain is valid for all integer values. For this product to be a perfect square, its square root must be an integer. Let's check if this is the case.
Since a and b are integers, so is their product ab. Therefore, we have shown that the square root of the product of two perfect squares, sqrt(A * B), is always an integer. Consequently, by definition, the product of two perfect squares is always a perfect square. Let's now can consider the quotient of two perfect squares.
A/B = a^2/b^2
Once more, A, B, a, and b are all positive integers, and b ≠ 0. For this quotient to be a perfect square, its square root must be an integer. Let's check if this is the case.
As we can see, the square root of the quotient of two perfect squares is equal to the quotient of two integers. However, this isn't always an integer. Consider the example shown below.
12=0.5
Therefore, since the square root of the quotient of two perfect squares is not always an integer, by definition, the quotient of two perfect squares is not always a perfect square.
