The square root · Code with Martin

A square root is a power backwards. How to find it, estimate it, and not get burned by negative numbers.

A power backwards

Last time: a square with side 7 cm, area 49 cm². Today the other way round. You know the area is 49 cm² and you ask: how long is the side?

You are looking for a number that, multiplied by itself, gives 49. That is 7. And that is exactly what a square root does: √49 = 7.

A square root is a power backwards. Power: from 7 make 49. Square root: from 49 make 7 again. The sign √ is called a radical and you read it “the square root of”.

That is why last lesson’s table of squares helps. If you know the squares, you know the square roots for free — you just read them the other way. 12² = 144, so √144 = 12.

And what about numbers that are not in the table? √50 does not come out neatly. You will learn to estimate it between two neighbours. That is a skill a calculator cannot replace — it checks whether the result makes sense.

Tereza is helping dad in the garden. They are planning a square bed and have 64 paving stones, each 1 dm². “How long will the side be?” dad asks. Tereza thinks: I need a number that times itself gives 64. She runs through the table: 7 · 7 = 49, too small. 8 · 8 = 64. “Eight stones along the side!” Dad counts the stones into a row and it fits. Tereza realises she just took a square root — only nobody at home calls it that.

💡

A square root of a “nice” number is the question: which number squared gives this? Do not look for a new method. Flip the table of squares you already know.

How to find √n

Step 1: try the table. √81 → which number squared gives 81? 9² = 81, so √81 = 9. Numbers like 1, 4, 9, 16, 25, 36… are called square numbers — their square root comes out whole.

Step 2: if the number is not in the table, estimate. √50: find the neighbours among square numbers. 49 < 50 < 64, so √50 is between 7 and 8. And because 50 sits right next to 49, it will be just a bit over 7 — about 7.1. Exactly it is 7.07…, a number with an endless decimal. An estimate is enough.

Step 3: check backwards with a square. You think √121 = 11? Check: 11² = 121. It fits.

And one important thing: you cannot take the square root of a negative number. √(−9) does not exist, because no number squared gives −9. Plus times plus is plus, minus times minus is plus too. The result of a square is never negative.

Example 1: √169

I am looking for a number that squared gives 169.

From last lesson’s table: 13² = 169. So √169 = 13.

Check backwards: 13 · 13 = 130 + 39 = 169. It fits.

If you did not have the table, you would narrow it with an estimate: 10² = 100 (too small), 15² = 225 (too big). It is between 10 and 15. The number 169 ends in nine — 3² and 7² end in nine. I try 13: hit.

Example 2: estimate √90

90 is not a square number. I find the neighbours:

9² = 81 and 10² = 100. We have 81 < 90 < 100, so √90 is between 9 and 10.

Where exactly? 90 is roughly in the middle of 81 and 100, maybe a little closer to 81. Estimate: about 9.5. The calculator says 9.49. A head estimate hit to a tenth — and that is exactly what you need so you can spot a typo on the calculator.

Example 3: the side of a square from its area

A square pitch has area 225 m². How long is the side?

Area of a square = side², so side = √225.

I look: 14² = 196, too small. 15² = 225. Hit. The side measures 15 m.

Check: 15 · 15 = 225 m². It fits. Notice the units: area is in m², the side in metres. The square root “took off” the two from the unit as well.

⚠️

√ applies only to what is under the radical. And most of all: a square root and dividing by two are not the same. √100 = 10, not 50. A square root asks “what squared?”, not “what is half?”.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. Start with ten. When it goes well, add more.

Take the square root. Every question has a whole-number answer — write only the number.

Practise the questions

On paper

Estimating is a muscle. You train it by hand and by comparing.

Take last lesson’s table of squares and rewrite it backwards: √1 = 1, √4 = 2, √9 = 3… up to √225 = 15.
Estimate without a calculator: √30, √75, √110. For each one write both neighbours (which square numbers it sits between) and your guess to one decimal place.
A square tablecloth has area 4 m². Calculate the side length and write a check by squaring.
Write in one sentence why √(−16) has no result.

Now you 💪

I find the square root of a square number through the question “what squared?”.
I can estimate √50 between 7 and 8 and I know why.
I know that a negative number under a square root has no result.

✅

Done when: You have the reverse table √1 to √225, three estimates with neighbours on paper, and a run of five correct in practice.

What to take from this lesson

√n is the number that squared gives n. A square root is a power backwards.
Square numbers (1, 4, 9, 16, 25…) have a whole-number square root.
Other square roots you estimate between two neighbours: 49 < 50 < 64, so √50 is between 7 and 8.
Always check the result backwards by squaring.
A negative number has no square root.