Finding Square Roots | Mathematics

Chapter 1 · A Square and A Cube

1.1 (continued) Finding Square Roots

Subtracting odd numbers tells you WHETHER a number is a perfect square. These two methods tell you WHAT the root actually is — one exact, one an estimate for when exact isn't possible by hand.

🎯 By the end of this you can…

✓Find an exact square root by pairing up prime factors
✓Spot instantly when a number is NOT a perfect square, from an unpaired prime
✓Estimate the square root of a large perfect square using tens and the last digit
✓Estimate the square root of a number that ISN'T a perfect square at all

💡 The idea

Method 1 — pair up the prime factors

A perfect square is some number $m$ multiplied by itself, so every prime factor of $m^{2}$ must appear an EVEN number of times — once for each copy of $m$ . Factorise into primes, pair them up two at a time, and take one factor from each pair.

🌳 √324 by pairing prime factors

⭐ 5 XP

324 \div 2 = 162

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📋 The method, in order

1
Break the number down into prime factors — smallest prime first, dividing again and again.
2
Group identical primes into pairs of two.
3
If every prime pairs up with no leftover, it's a perfect square — multiply one factor from each pair together for the root.
4
If even one prime is left without a partner, the number is NOT a perfect square at all.

🌳 √156 by pairing prime factors

⭐ 5 XP

156 \div 2 = 78

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⚠️ The classic trap

It's easy to stop the moment you see a repeated prime and assume it's "paired" without actually counting how many copies there are. $156 = 2 \times 2 \times 3 \times 13$ has 2 appearing twice (that pairs fine) but 3 and 13 each appear only ONCE — both unpaired, so 156 is not a perfect square, even though part of it looked promising.

💡 The idea

Method 2 — estimate by narrowing between tens

For a big number, factorising by hand is slow. Instead: find which two consecutive multiples of ten the root sits between, use the LAST DIGIT of the number to narrow it to just two candidates, then test one trial square to choose between them.

📏 Estimating √1936 — which tens, then which digit

⭐ 5 XP

4 0^{2} = 1600

and

5 0^{2} = 2500

, so

40 < 1936 < 50

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🤔 Why does the last digit of n narrow the root down to only TWO candidate digits, never just one?

💡 The idea

What if the number isn't a perfect square at all?

Then there's no EXACT whole-number root to find — only the nearest one. Bracket the number between two consecutive integers' squares and see which one it's closer to.

🔎 √250 isn't a whole number — which integer is it closest to?

⭐ 5 XP

1 5^{2} = 225

and

1 6^{2} = 256

, so

15 < 250 < 16

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✏️ Worked example

Akhil has a square cloth of area 125 cm² and wants the largest handkerchief with a whole-number side he can cut from it.

1
125 is not a perfect square — the nearest perfect squares are $1 1^{2} = 121$ and $1 2^{2} = 144$ .
2
A handkerchief of side 12 cm would need 144 cm² — more cloth than he has.
3
A handkerchief of side 11 cm needs only 121 cm² — this fits inside the 125 cm² he has.

✅ Answer: $11 cm$

🎮 Prime factors: perfect square, or not?

⭐ 0 XP🔥 streak 0

1156

a perfect square? If so, give

1156

2800

a perfect square?

What is the SMALLEST whole number that

9408

must be multiplied by to make it a perfect square?

Multiplying 9408 by that number gives a new perfect square — what is ITS square root?

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🏅

You finished this lesson!

Two solid methods now live in your toolkit: prime-factor pairing for an exact root, and tens-bracket estimation for anything too big to factorise by hand — or that isn't a perfect square at all.

Keep reading — this lesson marks itself done automatically.