Take the whole numbers and square them:
$0^2 = 0$
$1^2 = 1$
$2^2 = 4$
$3^2 = 9$
and so on.
The resulting numbers
$\,0, 1, 4, 9, 16, 25, 36, \ldots\,$ are called
perfect squares.
DEFINITION
perfect square
A number
$\,p\,$ is called a perfect square if and only if
there exists a whole number $\,n\,$ for which $\,p = n^2\,$.
In other words:
How do you get to be a perfect square?
Answer: By being equal to the square of some whole number.
(Recall that the whole numbers are
$\,0, 1, 2, 3, \ldots\,$)
In this exercise, you will decide if a given number is a perfect square.
The key is to rename the number (if possible) as a whole number, squared!
You may want to review this section first: Equal or Opposites?
EXAMPLES:
Question:
Is
$\,9\,$ a perfect square?
Solution:
Yes.
$\,9 = 3^2$
Question:
Is $\,7\,$ a perfect square?
Solution:
No. The number
$\,7\,$ can't be written as a whole number, squared.
Question:
Is
$\,17^2\,$ a perfect square?
Solution:
Yes. The number
$\,17\,$ is a whole number, so
$\,17^2\,$ is a whole number, squared.
Question:
Is
$\,17^4\,$ a perfect square?
Solution:
Yes. Rename as
$\,(17^2)^2\,$. The number $\,17^2\,$ is a whole number, so $\,(17^2)^2\,$ is a whole number, squared.
Question:
Is
$\,(-6)^2\,$ a perfect square?
Solution:
Yes. Rename as
$\,6^2\,$. The number $\,6\,$ is a whole number, so $\,6^2\,$ is a whole number, squared.
Question:
Is
$\,-6^2\,$ a perfect square?
Solution:
No. Recall that
$\,-6^2 = (-1)(6^2) = (-1)(36) = -36\,$. A perfect square can't be negative.
Be careful!
The numbers
$\,-6^2\,$ and $\,(-6)^2\,$ represent different orders of operation, and are different numbers!
Question:
Is $\,(-7)^{12}\,$ a perfect square?
Solution:
Yes. Rename:
$\,(-7)^{12} = 7^{12} = (7^6)^2\,$.
The number $\,7^6\,$ is a whole number, so $\,(7^6)^2\,$ is a whole number, squared.
Question:
Is
$\,-4\,$ a perfect square?
Solution:
No. A perfect square can't be negative.
Master the ideas from this section
by practicing the exercise at the bottom of this page.
When you're done practicing, move on to:
Writing Expressions in the form $\,A^2$