Perfect Square

Perfect Square

Any number that is the square of a rational number. For example, 0, 1, 4, 9, 16, 25, etc. are all perfect squares. So are $The fraction 1/25$ and $The fraction 9/4$ .

Key Formula

n = k^2

Where:

$n$ = The perfect square
$k$ = A rational number

Worked Example

Problem: Determine whether 36 is a perfect square.

Step 1: Ask: is there a rational number that, when multiplied by itself, gives 36?

k \times k = 36

Step 2: Test whole numbers:

6 \times 6 = 36

6^2 = 36

Step 3: Since 6 is a rational number and

6^2 = 36

, the number 36 is a perfect square.

Answer: Yes, 36 is a perfect square because

36 = 6^2

Why It Matters

Recognizing perfect squares lets you simplify square roots quickly — for instance,

\sqrt{49} = 7

requires no calculator. Perfect squares also appear when factoring quadratic expressions like

x^2 - 16 = (x-4)(x+4)

, and they are central to the technique of completing the square.

Common Mistakes

Mistake: Thinking perfect squares must be whole numbers only.

Correction: Any rational number squared produces a perfect square. For example,

\left(\frac{3}{4}\right)^2 = \frac{9}{16}

, so

\frac{9}{16}

is also a perfect square.

Related Terms

Rational Numbers — Perfect squares come from squaring these
Square Root — Inverse operation of squaring a number
Perfect Square Trinomial — Polynomial that factors as a binomial squared
Exponent — Perfect squares use an exponent of 2