﻿ Identifying Perfect Squares
IDENTIFYING PERFECT SQUARES
• PRACTICE (online exercises and printable worksheets)

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
Writing Expressions in the form $\,A^2$