A perfect square is a number that equals a whole number times itself, so its square root is a whole number. For example, 49 is a perfect square because 7 × 7 = 49 and √49 = 7. The first perfect squares are 0, 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100.
Perfect squares are the green dots: their square roots are whole numbers. Any number between two of them, like 50, has a square root that is not whole.
Formal definition and notation
An integer m is a perfect square if m = k2 for some integer k; equivalently, m ≥ 0 and √m is an integer. In prime factorization, m > 1 is a perfect square exactly when every prime appears to an even power, as in 144 = 24 × 32. In algebra, a perfect square trinomial is an expression of the form (a + b)2 = a2 + 2ab + b2.
Fractions such as 925 = (35)2 are sometimes called perfect squares too, because both parts are perfect squares.
Perfect square test
Perfect square test
m is a perfect square if √m is a whole number
m
the number you are testing (0 or more)
√196 = 14, so 196 is a perfect square. √200 ≈ 14.14, so 200 is not.
Try it: square a number
Interactive
The exponent is set to 2, so every result is a perfect square. Step the base from 1 to 12 and watch how fast the squares grow. Try a negative base too: the square is still positive.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Perfect Square examples, step by step
Example 1: Spot the perfect square
Warm-up
Which of 64, 90 and 169 are perfect squares?
82 = 64, so 64 is a perfect square.
92 = 81 and 102 = 100; 90 lies between them, so it is not.
132 = 169, so 169 is a perfect square.
Answer: 64 and 169 are perfect squares; 90 is not.
Example 2: Use prime factors
Core
Is 324 a perfect square? If so, find √324.
Factor: 324 = 4 × 81 = 22 × 34.
Both exponents (2 and 4) are even, so 324 is a perfect square.
Halve the exponents: √324 = 21 × 32 = 2 × 9 = 18.
Check: 18 × 18 = 324.
Answer: Yes; √324 = 18.
Example 3: Smallest multiplier
Test-style
What is the smallest whole number you can multiply 50 by to get a perfect square?
Factor: 50 = 2 × 52.
The 2 has an odd exponent; the 5 is already paired.
Multiply by 2: 50 × 2 = 100 = 22 × 52 = 102.
Answer: Multiply by 2, giving the perfect square 100.
Common mistakes
✗ Mistake√50 = 25, because 25 × 2 = 50.
✓ Instead√50 ≈ 7.07; 50 is not a perfect square. Use √50 = √25 × 2 = 5√2.
A square root multiplies by itself, not by 2, to make the number.
✗ Mistake-16 is a perfect square, because 4 times 4 is 16.
✓ Instead42 = 16 and (−4)2 = 16; nothing real squares to −16.
Any real number squared is 0 or positive.
✗ Mistake23 × 32 = 72 is a perfect square because it has a square factor.
✓ InsteadThe exponent of 2 is 3, which is odd, so 72 is not a perfect square: √72 ≈ 8.49.
Every prime exponent must be even, not just one of them.
The perfect square is the big number; the square root is the number that was squared. 81 is a perfect square, and 9 is its square root.
Perfect Squarevs.Perfect square trinomial
In algebra, x2 + 10x + 25 = (x + 5)2 is a perfect square trinomial: an expression, not a number, that is the square of a binomial.
Perfect Squarevs.Perfect cube
A perfect cube is a whole number cubed: 27 = 33. 64 is both a perfect square (82) and a perfect cube (43).
Quick check: 3 questions
Question 1 of 3
Which number is a perfect square?
Show answer and explanation
Answer: B.152 = 225. 200, 250 and 275 all end in 0 or 5 like 225, but none equals a whole number squared: √200 ≈ 14.1, √250 ≈ 15.8, √275 ≈ 16.6.
Question 2 of 3
Between which two whole numbers is √90?
Show answer and explanation
Answer: A.81 < 90 < 100, so 9 < √90 < 10. 8 and 9 would need a number between 64 and 81, 44 to 46 halves 90, and 10 to 11 needs 100 to 121.
Question 3 of 3
How many perfect squares are between 1 and 100, including both?
Show answer and explanation
Answer: D. They are 12 through 102: 1, 4, 9, …, 100, which is 10 numbers. 9 forgets 1 or 100, 11 adds 0, and 50 halves 100.
Perfect Square: frequently asked questions
How do you know if a number is a perfect square?
Take its square root. If the result is a whole number, it is a perfect square: √256 = 16. Without a calculator, check the last digit (perfect squares end in 0, 1, 4, 5, 6 or 9) and then use prime factors: the number is a perfect square only if every prime appears an even number of times.
Is 0 a perfect square?
Yes. 0 = 02, and 0 is a whole number, so 0 counts as a perfect square, and √0 = 0. Many lists start at 1 simply because 0 is an unusual example, but by the definition “the square of a whole number” 0 qualifies.
Why do perfect squares matter on tests?
They make square roots exact. On the SAT, ACT and GED you simplify radicals by pulling out perfect squares, √48 = √16 × 3 = 4√3, estimate roots like √70 as just over 8, and spot Pythagorean triples such as 9 + 16 = 25.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 8.EE.A.2, CCSS 6.EE.A.1
Cite this entry
Effortless Math Team. “Perfect Square.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/perfect-square/
Flashcard
TermPerfect Square√m is a whole number
DefinitionA number that is a whole number squared, so its square root is whole. 49 is one (72); 50 is not (√50 ≈ 7.07).
Square RootA square root of a number is a value that, multiplied by itself, gives that number. The square root of 36 is 6 because 6 × 6 = 36. Every positive number has two square roots, such as 6 and −6, but the radical sign √ means the principal square root, which is the non-negative one.Number System · Grade 8, Algebra 1Prime FactorizationPrime factorization is writing a whole number as a product of prime numbers only, such as 72 = 2 × 2 × 2 × 3 × 3 = 23 × 32. Find it with a factor tree or repeated division. Every whole number greater than 1 has exactly one prime factorization, apart from the order of the factors.Number System · Grade 4, Grade 5, Grade 6PowerA power is a number written as a base with an exponent, such as 53, or the value that results, 53 = 5 × 5 × 5 = 125. We say 125 is the third power of 5, or “5 to the third power”. The second power is called the square and the third power the cube.Number System · Grade 5, Grade 6, Grade 7TrinomialA trinomial is a polynomial with exactly three terms, such as x2 + 5x + 6 or 2a − 3b + 7. The terms are joined by addition or subtraction and must be unlike terms, so nothing combines. The most common kind is the quadratic trinomial ax2 + bx + c, which students learn to factor into two binomial factors.Algebra · Algebra 1, Algebra 2Whole NumberA whole number is 0 or a positive counting number: 0, 1, 2, 3, and so on. Whole numbers do not include fractions, decimals, or negative numbers, so 12 is a whole number but 8.5, 34 and −3 are not. Every whole number is also an integer.Number System · Grade 3, Grade 4, Grade 5SquareA square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s2, and each diagonal is s√2.Geometry · Kindergarten, Grade 2, Grade 3ExponentAn exponent tells how many times to use a base as a factor. It is the small raised number to the upper right of the base: in 53, the base is 5 and the exponent is 3, so 53 = 5 × 5 × 5 = 125. Any nonzero number with an exponent of 0 equals 1.Algebra · Grade 6, Grade 7, Grade 8NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2RadicalA radical is an expression that uses the root symbol √, such as √25 or 3√x. The number under the symbol is the radicand, and the small number in the notch is the index, which tells which root to take. With no index, it is a square root: √25 = 5.Number System · Grade 8, Algebra 1, Algebra 2Negative NumberA negative number is a number less than zero, written with a minus sign, such as −3, −0.5 or −23. On a number line, negative numbers sit to the left of 0. They describe amounts below a starting point: temperatures below zero, depths below sea level, or money owed. Zero itself is neither negative nor positive.Number System · Grade 6, Grade 7Real NumberA real number is any number that can be placed on the number line. The real numbers are all the rational numbers (fractions, integers, terminating and repeating decimals) together with all the irrational numbers (such as √2 and π). Square roots of negative numbers, like √−4, are not real.Number System · Grade 8, Algebra 1BinomialA binomial is a polynomial with exactly two terms, joined by a plus or minus sign. Examples are x + 7, 3m − 4 and a2 − 9. Combine like terms before counting: 3x + 5x is 8x, one term. A polynomial with one term is a monomial; three terms make a trinomial.Algebra · Algebra 1, Algebra 2Even NumberAn even number is an integer that can be divided by 2 with no remainder, so it can be split into two equal groups or into pairs. Even numbers end in 0, 2, 4, 6 or 8, such as 14, 236 and −10. Zero is even. Every even number can be written as 2n for some integer n.Number System · Grade 2, Grade 3, Grade 4MultiplicationMultiplication is the operation that combines equal groups to find a total. The fact 4 × 6 = 24 means 4 groups of 6, or 6 + 6 + 6 + 6. The numbers you multiply are called factors, and the result is the product. Order does not matter: 6 × 4 is also 24.Operations · Grade 2, Grade 3, Grade 4Irrational NumberAn irrational number cannot be written as a fraction of two integers. Its decimal form does not end and does not repeat in a pattern. Examples are √2 = 1.41421… and π = 3.14159…. The square root of any whole number that is not a perfect square is irrational.Number System · Grade 8, Algebra 1Pythagorean TheoremThe Pythagorean theorem says that in a right triangle, a2 + b2 = c2. The legs are a and b, the two sides that form the right angle. The hypotenuse, c, is the longest side across from the right angle. Legs of 3 and 4 give 9 + 16 = 25, so the hypotenuse is 5.Geometry · Grade 8, Algebra 1Completing the SquareCompleting the square is a method that rewrites x2 + bx as a perfect square by adding (b2)2: x2 + bx + (b2)2 = (x + b2)2. It is used to solve quadratic equations, to write a quadratic in vertex form, and to derive the quadratic formula.Algebra · Algebra 1, Algebra 2, PrecalculusDifference of SquaresA difference of squares is an expression of the form a2 − b2: one perfect square subtracted from another. It always factors as (a + b)(a − b). For example, x2 − 49 = (x + 7)(x − 7). A sum of squares, a2 + b2, does not factor over the real numbers.Algebra · Algebra 1, Algebra 2Square NumberA square number is the result of multiplying a whole number by itself. The first square numbers are 1, 4, 9, 16, 25 and 36, because 1 × 1 = 1, 2 × 2 = 4, 3 × 3 = 9 and so on. That many dots fill a square exactly, like 16 dots in a 4 by 4 square.Operations · Grade 3, Grade 4, Grade 5Cube RootThe cube root of a number is the value that, used as a factor three times, gives that number. It is written 3√x. Because 4 × 4 × 4 = 64, 3√64 = 4. Unlike square roots, cube roots of negative numbers are real numbers: 3√−27 = −3.Number System · Grade 8, Algebra 1, Algebra 2