Identifying Perfect Squares

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 $\,A^2$