The resulting numbers
$\,0,\, 1,\, 4,\, 9,\, 16,\, 25,\, 36,\, \ldots\,$ are called
perfect squares.
DEFINITIONperfect square
A number
$\,p\,$ is called a perfect squareif and only ifthere 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.
Identifying Perfect Squares