Perfect Squares and Square Roots

What is a perfect square?

A perfect square is a whole number multiplied by itself. 7 × 7 = 49, so 49 is a perfect square. We write it 7² and read it seven squared.

Two readings of the same square:

• Side length. Drag to side 7. The square is 7 × 7.
• Area. That square contains 49 unit tiles. 49 is the area.

So 7² = 49 says both things: 7 multiplied by itself, and the number of tiles in a 7 × 7 square.

In 7², the 7 is the base and the 2 is the exponent. The exponent counts how many copies of the base get multiplied together, not what to multiply by. So 5² = 5 × 5 = 25, never 5 × 2 = 10.

The square root reverses the question: given the area, what's the side? $\sqrt{49} = 7$ because a square of area 49 has side 7.

Which numbers are perfect squares?

Most numbers aren't perfect squares. Between 49 = 7² and 64 = 8² sit 50, 51, 52, …, 63: fourteen integers, none of them perfect. The perfect squares thin out as you go: the gap between consecutive ones grows by 2 every step.

A non-perfect square sits between two perfect ones. 50 is between 49 = 7² and 64 = 8², so $\sqrt{50}$ is between 7 and 8. Estimating those in-between roots comes later.

Worksheet

Try these to put the idea into practice. The goal is recall for 1² through 12².

The curriculum expects fluent recall of 1² through 12². Drill them until they're automatic.

Going further

Perfect cubes come next. 5³ = 125 is the count of unit cubes in a 5 × 5 × 5 stack, and

$\sqrt[3]{125} = 5$ is the side length. The exponent-as-factor trap reappears there as 5³ = 15, and the base / exponent vocabulary applies the same way.

Square roots of non-perfect squares, like

$\sqrt{50}$, sit between two whole numbers. $\sqrt{50}$ is between 7 and 8, closer to 7. Estimating those values is the next step.