IDENTIFYING PERFECT SQUARES

by Dr. Carol JVF Burns (website creator)

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- PRACTICE (online exercises and printable worksheets)
- Click here for a printable pdf of this lesson

Take the whole numbers and square them:

$0^2 = 0$

$1^2 = 1$

$2^2 = 4$

$3^2 = 9$

and so on.$1^2 = 1$

$2^2 = 4$

$3^2 = 9$

The resulting numbers $\,0, 1, 4, 9, 16, 25, 36, \ldots\,$ are called

DEFINITION
perfect square

A number
$\,p\,$ is called a *perfect square*
if and only if

there exists a whole number $\,n\,$ for which $\,p = n^2\,$.

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!

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.

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 $\,A^2$

by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:

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