﻿ Identifying Perfect Squares

# Identifying Perfect Squares

Take the whole numbers and square them:

$$\begin{gather} \cssId{s2}{0^2 = 0}\cr \cssId{s3}{1^2 = 1}\cr \cssId{s4}{2^2 = 4}\cr \cssId{s5}{3^2 = 9}\cr \end{gather}$$

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 $\,\color{red}{17}\,$ is a whole number, so $\,\color{red}{17}^2\,$ is a whole number, squared.
Question: Is $\,17^4\,$ a perfect square?
Solution: Yes.  Rename as $\,(17^2)^2\,.$ The number $\,\color{red}{17^2}\,$ is a whole number, so $\,(\color{red}{17^2})^2\,$ is a whole number, squared.
Question: Is $\,(-6)^2\,$ a perfect square?
Solution: Yes.  Rename as $\,6^2\,.$ The number $\,\color{red}{6}\,$ is a whole number, so $\,\color{red}{6}^2\,$ is a whole number, squared.
Question: Is $\,-6^2\,$ a perfect square?
Solution: No.  Recall that: $$\,\cssId{s53}{-6^2 = (-1)(6^2) = (-1)(36) = -36}$$ 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: $$\cssId{s62}{(-7)^{12} = 7^{12} = (7^6)^2}$$ The number $\,\color{red}{7^6}\,$ is a whole number, so $\,(\color{red}{7^6})^2\,$ is a whole number, squared.
Question: Is $\,-4\,$ a perfect square?
Solution: No.  A perfect square can't be negative.

YES
NO