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Recognize perfect squares

A perfect square is a nonnegative integer that can be written as n2n^2 for a whole number nn, such as 0,1,4,9,…,2250,1,4,9,\ldots,225; its principal square root is therefore a whole number. Recognizing familiar perfect squares and distinguishing them from non-squares supports interpreting square-root notation and solving measurement and geometric problems, without extending to large-number algorithms, irrational-root approximation, or more advanced algebraic generalizations.

Detailed Explanation: Recognize perfect squares

A perfect square is a whole number that can be written as a whole number multiplied by itself:

n2=n×nn^2=n\times n

For example, (9=32=3×3)(9=3^2=3\times3), so 99 is a perfect square. The principal square root of a perfect square is a whole number:

9=3\sqrt{9}=3

To recognize a perfect square, compare the number with familiar squares:

02=012=122=432=942=1652=2562=3672=4982=6492=81102=100112=121122=144132=169142=196152=225\begin{aligned} 0^2&=0 & 1^2&=1 & 2^2&=4 & 3^2&=9\\ 4^2&=16 & 5^2&=25 & 6^2&=36 & 7^2&=49\\ 8^2&=64 & 9^2&=81 & 10^2&=100 & 11^2&=121\\ 12^2&=144 & 13^2&=169 & 14^2&=196 & 15^2&=225 \end{aligned}

Example: Is (144)(144) a perfect square?

  1. Look for a whole number that multiplied by itself equals (144)(144).

  2. Recall that (122=12×12)(12^2=12\times12).

  3. Calculate:

12×12=14412\times12=144

  1. Therefore,

144=122144=12^2

So, (144)(144) is a perfect square, and its principal square root is:

144=12\sqrt{144}=12

If no whole number squared equals a number, then the number is not a perfect square.

Learn by doing: Recognize perfect squares

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    Square Roots of Perfect Squares

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