Perfect Square
In short: A perfect square is a number you get by multiplying a whole number by itself, such as 49 = 7 × 7, so its square root is itself a whole number.
8th GradeNumbers and Operations8.EE.A.2
What is Perfect Square?
Arrange dots in a square array with the same number of rows and columns and the total is a perfect square: 1, 4, 9, 16, 25, 36 and so on. Formally, \(n\) is a perfect square when \(n = k^2\) for some integer \(k\).
The list grows by successive odd numbers: \(1\), \(1+3=4\), \(4+5=9\), \(9+7=16\). Such numbers can only end in 0, 1, 4, 5, 6 or 9, which gives a quick way to rule out numbers. Their square roots are whole numbers, so they are rational.
How to spot it
- A problem asks whether a number such as 169 or 200 can be written as an integer times itself.
- A square root must be simplified or estimated and the nearest perfect squares are needed as anchors.
- A square tile pattern or dot array uses the same count in each row and column.
- Answers to a root question are whole numbers rather than long decimals.
The method
- Estimate a whole number whose square is near the target, using squares you know.
- Square the candidate and compare it with the target; adjust up or down by one.
- If you land exactly on the target, it qualifies; if the target falls between two consecutive squares, it is not.
- Use the last-digit rule as a shortcut: a number ending in 2, 3, 7 or 8 is never a perfect square.
Perfect Square: solved example
Problem
Is 169 a perfect square? Is 200?
Solution
Try \(13 \times 13 = 169\), so 169 is a perfect square with root 13.
For 200, \(14^2 = 196\) and \(15^2 = 225\), so 200 lies strictly between two consecutive squares. 169 qualifies; 200 does not.
Try it
Question 1
List the perfect squares between 50 and 150.
Show the answer
\(8^2 = 64\), \(9^2 = 81\), \(10^2 = 100\), \(11^2 = 121\), \(12^2 = 144\): so 64, 81, 100, 121 and 144.
Question 2
Without computing, explain why 2,537 cannot be the square of a whole number.
Show the answer
It ends in 7, and no whole number squared ends in 7.
Perfect Square vs. Square Root
This kind of number, such as 144, sits on the list of squares of whole numbers. A square root is an operation, or its answer, that goes from the square back to the original. Each of those has a whole-number root; most other numbers have a root that is only approximated.
Keep going
Lessons
Worksheets and quizzes
- Square Roots and Cube Roots – Worksheet
- Square Roots and Cube Roots – Quiz
- Exponents and Order of Operations – Worksheet
- Exponents and Order of Operations – Quiz
Related words
Whole dictionary: A–Z · 8th Grade · Numbers and Operations