IDENTIFYING PERFECT SQUARES

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Take the whole numbers and square them:

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$0^2 = 0$

$1^2 = 1$

$2^2 = 4$

$3^2 = 9$

and so on.

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

DEFINITION
perfect square

In other words:
A number
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$\,p\,$ 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 $\,p = n^2\,$.

How do you get to be a

Answer: By being equal to the square of some whole number.

(Recall that the

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
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$\,9\,$ a perfect square?

Solution:
Yes.
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$\,9 = 3^2$

Question:
Is $\,7\,$ a perfect square?

Solution:
No. The number
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$\,7\,$ can't be written as a whole number, squared.

Question:
Is
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$\,17^2\,$ a perfect square?

Solution:
Yes. The number
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$\,17\,$ is a whole number, so
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$\,17^2\,$ is a whole number, squared.

Question:
Is
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$\,17^4\,$ a perfect square?

Solution:
Yes. Rename as
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$\,(17^2)^2\,$. The number $\,17^2\,$ is a whole number, so $\,(17^2)^2\,$ is a whole number, squared.

Question:
Is
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$\,(-6)^2\,$ a perfect square?

Solution:
Yes. Rename as
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$\,6^2\,$. The number $\,6\,$ is a whole number, so $\,6^2\,$ is a whole number, squared.

Question:
Is
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$\,-6^2\,$ a perfect square?

Solution:
No. Recall that
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$\,-6^2 = (-1)(6^2) = (-1)(36) = -36\,$. A perfect square can't be negative.

Be careful!

The numbers [beautiful math coming... please be patient] $\,-6^2\,$ and $\,(-6)^2\,$ represent different orders of operation, and are different numbers!

Be careful!

The numbers [beautiful math coming... please be patient] $\,-6^2\,$ and $\,(-6)^2\,$ represent different orders of operation, and are different numbers!

Question:
Is $\,(-7)^{12}\,$ a perfect square?

Solution:
Yes. Rename:
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$\,(-7)^{12} = 7^{12} = (7^6)^2\,$.

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
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$\,-4\,$ 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 $\,A^2$