Perfect Square Integers

Lesson Features »

Lesson Priority: Normal

  • Recall what we know so far about square roots from Pre-Algebra
  • Remember what rational vs irrational numbers are, and know when square roots are one or the other
  • Know what perfect square integers are
  • Be able to quickly generate and identify perfect square integers
  • Be familiar with the square roots of the common perfect square integers
Lesson Description

Square roots are a common operation that we probably didn't need much before now. Here we'll make sure we understand exactly what roots are, and that when a square root of a number is rational, it is because the number was what we call a "perfect square". In addition to helping us better understand what square roots are, this lesson will help us better understand where perfect square integers come from, and how we can quickly generate a list of them.

Practice Problems

Practice problems and worksheet coming soon!


Square Roots So Far

In Pre-Algebra we get a small introduction » to what square roots are.Let's face it - square roots do not have a reputation for being students' favorite part of Algebra, because they seem so fundamentally different from arithmetic-style operations we learn and seem to have their own set of rules.The best thing you can do is to help ground yourself and remember conceptually what square roots are.
Pro Tip
In Pre-Algebra, we said that taking a square root is the opposite operation of squaring a number. If you have trouble understanding the nature of square roots, I strongly recommend using this definition to help you.
We should also remember that certain whole numbers have square roots that are also whole numbers. If the square root of an integer is not also an integer, than it is irrational.

Rational vs. Irrational

Though this is something that gets discussed on and off, let's quickly remember what rational and irrational mean.A rational number is any number that can be expressed as a fraction of whole numbers. This includes whole numbers themselves, e.g. $5$ is a rational number because$$5 = \frac{5}{1}$$Irrational numbers on the other hand cannot be expressed as a fraction. When equipped with a calculator, the tell-tale sign that a result is irrational is that the decimal does not have a pattern in it.

Knowing Perfect Squares

As we said, any real number is either rational or irrational. When you encounter irrational numbers in mathematics, you'll most commonly see square roots (or other roots) involved, because the square root of any integer is irrational if that integer is not a perfect square.
I Used To Know That!
In Pre-Algebra, we defined perfect square integers as the result of squaring any integer. These are the only integers that have rational square roots.
There are two things you need to know to move forward with confidence:
  • How to generate and recognize whether an integer is a perfect square or not
  • What the square roots of the first $25$ or so integers are
Neither of these tasks is accomplished with anything other than exposure and practice, but there is a way to generate the list by-hand when you need it.Generate Perfect SquaresIf you are trying to take the square root of a number and you are unsure if it is a perfect square integer, simply re-generate the list of perfect square integers by listing the integers in order and squaring them. This is a task you can quickly do on quizzes, vertically in the margin of your paper:$$1^2 = 1$$$$2^2 = 4$$$$3^2 = 9$$$$4^2 = 16$$$$5^2 = 25$$$$\dots$$$$20^2 = 400$$Then, if your number appears in the list, then you not only know it's a perfect square integer that has a nice square root, but you also know what that square root is! And, if your number gets skipped over, then you know it is not a perfect square.

Put It To The Test

Example 1Identify which of the following numbers has a rational square root, and if so, state that square root.$32$, $49$, $56$, $72$, $99$, $121$, $169$, $200$, and $289$
Show solution
$\blacktriangleright$ If you are familiar with the list of perfect squares, this could be a memory game for you. Otherwise, just generate the complete list of perfect square integers until you reach a large enough number that you know you've checked everything on this list.You should have determined that $49$, $121$, $169$, and $289$ are each perfect squares, and their respective square roots are $7$, $11$, $13$, and $17$.
Lesson Takeaways
  • Continue to gain comfortability with what square roots are
  • Remember that integers either have integer square roots or irrational ones
  • Know which integers do and do not have rational square roots
  • Know how to generate and recognize the integers that have integer square roots

Lesson Metrics

At the top of the lesson, you can see the lesson title, as well as the DNA of Math path to see how it fits into the big picture. You can also see the description of what will be covered in the lesson, the specific objectives that the lesson will cover, and links to the section's practice problems (if available).

Key Lesson Sections

Headlines - Every lesson is subdivided into mini sections that help you go from "no clue" to "pro, dude". If you keep getting lost skimming the lesson, start from scratch, read through, and you'll be set straight super fast.

Examples - there is no better way to learn than by doing. Some examples are instructional, while others are elective (such examples have their solutions hidden).

Perils and Pitfalls - common mistakes to avoid.

Mister Math Makes it Mean - Here's where I show you how teachers like to pour salt in your exam wounds, when applicable. Certain concepts have common ways in which teachers seek to sink your ship, and I've seen it all!

Put It To The Test - Shows you examples of the most common ways in which the concept is tested. Just knowing the concept is a great first step, but understanding the variation in how a concept can be tested will help you maximize your grades!

Lesson Takeaways - A laundry list of things you should be able to do at lesson end. Make sure you have a lock on the whole list!

Special Notes

Definitions and Theorems: These are important rules that govern how a particular topic works. Some of the more important ones will be used again in future lessons, implicitly or explicitly.

Pro-Tip: Knowing these will make your life easier.

Remember! - Remember notes need to be in your head at the peril of losing points on tests.

You Should Know - Somewhat elective information that may give you a broader understanding.

Warning! - Something you should be careful about.

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