Key takeaway: A perfect square is a whole number that equals an integer multiplied by itself, such as 49, which is 7 times 7.
Grade 88.EE.A.2Numbers and Operations
Definition of Perfect Square
A perfect square is what you get when you multiply a whole number by itself. The name comes from arranging dots or tiles: 16 tiles can make a 4-by-4 square with none left over, but 18 tiles cannot make a square.
Knowing the first fifteen or so by heart speeds up work with roots, because their roots are whole numbers. A reliable test uses prime factors. Break the number into primes; if every prime appears an even number of times , the test passes. The sequence starts 1, 4, 9, 16, 25, and the gaps between neighbors keep growing by 2: 3, 5, 7, 9.
Formula and notation
A number \(N\) belongs to this family when
\[N = n^2 \quad \text{for some integer } n\]
Squaring 1 through 15 gives values from \(1^2 = 1\) up to \(15^2 = 225\).
Key facts
- Its last digit is never 2, 3, 7 or 8.
- Every prime in its factorization appears an even number of times .
- Its square root is itself a whole number.
- The difference between consecutive ones is the next odd number.
Perfect Square example
Use prime factorization to decide which of 200 and 1,296 is the square of a whole number, and give its square root.
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\(200 = 2^3 \cdot 5^2\). The prime 2 appears 3 times , an odd number, so 200 fails.
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\(1296 = 2^4 \cdot 3^4\). Both exponents are even, so 1,296 passes.
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Halve the exponents: \(2^2 \cdot 3^2 = 4 \cdot 9 = 36\). Check: \(36 \times 36 = 1296\).
1,296 is a perfect square, with \(\sqrt{1296} = 36\); 200 is not.
Common mistake
Watch out
Omari reasons that 18 is a perfect square, because \(18 = 3 \times 6\), and both of those are whole numbers.
Why it is wrong
A perfect square needs the same whole number used twice. Here \(18 = 2 \cdot 3^2\), and the prime 2 appears only once. The nearby squares are \(16 = 4^2\) and \(25 = 5^2\), with 18 stuck in between.
Related terms and lessons
Browse the glossary: A–Z list · Grade 8 vocabulary · Numbers and Operations