Finding the side of a square

A square has an area of 49 cm². How long is each side? We need two equal lengths whose product is 49. Since 7 × 7 = 49, each side is 7 cm long.

Finding a square root means working backwards from a square to the number that was multiplied by itself. We write √49 = 7, read as “the square root of 49 is 7”. The root symbol gives the non-negative root: a positive number, or zero.

To check a square root, multiply your answer by itself. Halving the area won't find the side: the two equal numbers must multiply to make the area, not add to it.

Using the squares you know

Every square fact also gives us a root fact. Knowing that 12 × 12 = 144 tells us that √144 = 12. The operation has changed direction, but the numbers belong together.

Not every whole number is a square number. For example, 50 lies between 49 and 64, so √50 lies between 7 and 8. It still has a square root; that root just isn't a whole number.

Negative numbers can also give positive squares when multiplied by themselves. We'll separate those two ideas in the next lesson.

Square roots of decimals

The same check works with decimals. Since 0.2 × 0.2 = 0.04, we have √0.04 = 0.2.

Making each factor ten times smaller makes the product a hundred times smaller. Reading backwards, √1.44 = 1.2, not 0.12: squaring 0.12 gives 0.0144 instead.

For the same reason, √0.0016 = 0.04: four hundredths multiplied by four hundredths makes sixteen ten-thousandths. Checking by multiplication keeps the decimal point in the right place.

When the root is larger

We might expect taking a square root to make a number smaller. But √0.25 = 0.5, and 0.5 is larger than 0.25.

Multiplying 0.5 by 0.5 means taking half of 0.5, giving 0.25. Taking the square root reverses that change, from 0.25 back to 0.5. This happens for every number between 0 and 1: its square root is larger than the number itself.