MathIt BlogSquare roots explained: what they are and how to find them
What is a square root and how do you find it?
A square root of a number is the value that gives the number when multiplied by itself, so √49 = 7 because 7 × 7 = 49. To find one, recall it if the number is a perfect square. Otherwise, place the number between two perfect squares, estimate, and check by squaring. Prime factorization and a calculator also work.
Key takeaways
- A square root undoes squaring, so √49 = 7 because 7 × 7 = 49.
- The first 15 square numbers run from 1 to 225, and knowing them by heart makes most square roots a matter of recall.
- A square root of a number that is not a perfect square lies between two whole numbers, so √50 is between 7 and 8 and is about 7.07.
- The square root of a sum is not the sum of the square roots: √(9 + 16) = √25 = 5, while √9 + √16 = 7.
- The √ sign means the positive square root, so √49 = 7, and a negative number such as −4 has no real square root.
Your seventh grader is stuck on √50, and the calculator answer 7.0710678 looks like it came from nowhere. A square root sounds like a new operation, yet it is squaring run backward, and square numbers from the times tables hold almost every answer. Once that link is clear, you can find a square root by hand and check it in seconds.
What is a square root?
A square root of a number is the value that gives that number when you multiply it by itself. √49 = 7 because 7 × 7 = 49, and √100 = 10 because 10 × 10 = 100. The sign √ is called the radical sign, and you read √49 as "the square root of 49".
The OpenStax Prealgebra textbook gives the definition in one line: "A number whose square is m is called a square root of m." Squaring and taking a square root undo each other the way multiplication and division do. The guide to exponents explained covers the squaring half.
A picture makes the link concrete. A square made of 49 tiles has 7 tiles along each side, so the square root of an area is the length of the side.
Square numbers: the first 15 perfect squares
Square numbers, also called perfect squares, are the results of multiplying a whole number by itself. Their square roots are whole numbers, so every square number is also a square root fact.
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| n² | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 | 169 | 196 | 225 |
England's national curriculum for mathematics asks Year 5 pupils to "recognise and use square numbers and cube numbers, and the notation for squared (²) and cubed (³)". The first ten rows come straight from the times tables, which is why the post on how to learn times tables fast pays off here too.
How to find a square root, step by step
A square root is found in four steps: check for a perfect square, place the number between two perfect squares, estimate, and check by squaring. Take √50 as the worked example.
- Ask whether it is a perfect square. If the number appears among the square numbers, recall the root: √144 = 12. The number 50 is not on the list, so move on.
- Find the two perfect squares around it. 50 sits between 49 = 7² and 64 = 8², so √50 is between 7 and 8.
- Estimate from the distances. 50 is 1 above 49 and 14 below 64, so √50 is much nearer 7. The gap from 49 to 64 is 15, so a good first guess is 7 + 1⁄15, about 7.07.
- Check by squaring. 7.07 × 7.07 = 49.9849, which is close to 50. A calculator gives √50 ≈ 7.0711, so the estimate is right to two decimal places.
The same four steps work for any whole number. For √30, the neighbors are 25 and 36, so the root lies between 5 and 6, a little below the middle, and 5.5 × 5.5 = 30.25 confirms it.
The part people get wrong: roots of sums
The square root of a sum is not the sum of the square roots. √(9 + 16) = √25 = 5, while √9 + √16 = 3 + 4 = 7. Work out everything under the radical sign first, then take the root.
OpenStax makes the same point with a worked pair: "In ⓐ, we took each square root first and then added them. In ⓑ, we added under the radical sign first and then found the square root." Multiplication does split, so √(4 × 9) = √36 = 6 and √4 × √9 = 2 × 3 = 6.
Four ways to find a square root side by side
The four ways to find a square root are recall, estimate-and-check, prime factorization and a calculator. Each one suits a different number and a different moment in school.
| Method | How it works | Best for | Limits |
|---|---|---|---|
| Recall | Match the number to a known square: √81 = 9 | Perfect squares up to 225 or 400 | Needs the squares memorized |
| Estimate and check | Place between two squares, guess, square the guess | √50 ≈ 7.07, quick checks, tests without a calculator | Gives an approximation, usually 1–2 decimal places |
| Prime factorization | Split into primes, halve each exponent: √144 = √(2⁴ × 3²) = 2² × 3 = 12 | Large perfect squares such as 324 or 576 | Slow for big numbers; no whole-number answer for non-squares |
| Calculator | Press √ or type =SQRT(50) in a spreadsheet | Any number, any precision | Hides whether the answer is sensible |
Prime factorization works because a perfect square has every prime an even number of times. 324 = 2² × 3⁴, so √324 = 2 × 3² = 18, and 18 × 18 = 324 confirms it. The post on prime numbers explained shows how to build the factor tree, and its divisibility test stops at the square root of the number for the same reason.
Which method for which grade
The grade decides which square-root method comes first. Recall comes first, because square numbers arrive in grades 5 and 6 with the times tables and the ² notation.
In the US, square roots arrive in grade 8: Common Core standard 8.EE.A.2 asks students to "evaluate square roots of small perfect squares" and to "know that √2 is irrational".
Standard 8.NS.A.2 adds estimation: "by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5". Prime factorization fits grades 7 and 8, and the calculator comes last, as a check on an estimate.
Practical tips so square roots stick
Square roots stick when the square numbers are automatic, so learn them to 15², and to 20² if your child is heading for algebra. Five minutes a day on the square numbers game gets 13² = 169 and 14² = 196 to come back without working.
Use the last digit as a filter. A perfect square ends only in 0, 1, 4, 5, 6 or 9, so 2, 3, 7 and 8 never end one.
The last-digit rule rules out 47, 83 and 572 in one glance. The rule only filters: 26 ends in 6 and is still not a square.
Link roots to mental math. √2,500 = 50 because 50 × 50 = 2,500, and √1,600 = 40 for the same reason. The guide to mental math tricks that work has more shortcuts of that kind.
Play free in your browserTimes TablesGrade 3+A practice rule: always square the answer
Squaring the answer is the one habit that catches every wrong square root. If √196 came out as 13, then 13 × 13 = 169 shows the slip, and 14 × 14 = 196 shows the fix. Ask for the check on every homework question until your child does it without being asked.
How to check progress with square roots
Five signals show whether square roots make sense, and each takes under a minute.
- Recall: Ask for √121, √169 and √225. The answers 11, 13 and 15 should come in under three seconds each.
- Placement: Ask between which two whole numbers √90 lies. The answer is 9 and 10, because 81 < 90 < 100.
- Estimate: Ask for √65 to one decimal place. About 8.1 is right, because 65 is 1 above 64.
- Sums under the sign: Give √(36 + 64). The answer is 10, and 14 means the roots were taken first.
- The reason: Ask why √0.25 = 0.5. "Because 0.5 × 0.5 = 0.25" shows the idea; "because it halves" does not.
A five-minute test
A five-minute test mixes one item of each type and is repeated twice a week with new numbers. Mark the check as well as the answer: a student who writes √50 ≈ 7.07 and squares it to 49.98 understands roots. The math game for 6th graders page gathers practice at the grade where squares and exponents meet.
Common square root mistakes and edge cases
Five mistakes cause most wrong square roots, and each has a one-line fix.
- √16 = 8: The number was halved. Square the answer: 8 × 8 = 64, so the root is 4.
- √(9 + 16) = 7: The roots were taken before the addition. Add first: √25 = 5.
- √50 = 25: The same halving slip on a larger number. Place 50 between 49 and 64 instead.
- √2 = 1.414 exactly: √2 is irrational, so write √2 ≈ 1.414; 1.414 × 1.414 = 1.999396.
- √49 = ±7: The radical sign gives one answer. Write √49 = 7, and −√49 = −7 for the negative root.
Negative numbers need care. Both 7 × 7 and (−7) × (−7) equal 49, so 49 has two square roots, and the √ sign picks the positive one, called the principal square root. OpenStax states it plainly: "The radical sign, √, stands for the positive square root."
Square roots of negative numbers
A negative number has no real square root, because any real number times itself is zero or positive. OpenStax says that "there is no real number whose square is −169", so √−4 has no real answer. The post on negative numbers explained covers why a negative times a negative is positive, which is the reason behind the rule.
Your square roots checklist and next steps
- Know the squares: 1² to 15² by heart, with 225 as the last.
- Place the number: find the perfect squares below and above it.
- Estimate by distance: nearer the lower square means nearer the lower root.
- Work under the sign first: √(a + b) is never split into √a + √b.
- Square the answer: the check takes ten seconds and catches every slip.
England's curriculum takes roots further at Key Stage 3, where pupils "distinguish between exact representations of roots and their decimal approximations". That is the line between √2 and 1.414, and the five habits above prepare for it.
MathIt is a free app for iOS and Android with about 90 game types, adaptive difficulty and daily practice, so square numbers get harder only after the easier ones come out right. Twenty-two of its games, square numbers among them, play free in a browser with no sign-up.
Practise it in the MathIt app
About 90 math games with adaptive difficulty, daily practice and 2–5 player battles — free on iOS and Android.
Frequently asked questions
What is the square root of 64?
The square root of 64 is 8, because 8 × 8 = 64. The number 64 is a perfect square, the eighth square number. The number −8 also gives 64 when squared, but the √ sign means the positive root, so √64 = 8.
What is the square root of 2?
The square root of 2 is about 1.414, because 1.414 × 1.414 = 1.999396. The square root of 2 is irrational: its decimal never ends and never repeats, so any written value is an approximation. Common Core standard 8.EE.A.2 asks eighth graders to know that √2 is irrational.
What is the square root of 16?
The square root of 16 is 4, because 4 × 4 = 16. A common slip is to answer 8, which is 16 halved. Halving and taking a square root only agree for 0 and 4, where 4 ÷ 2 = 2 and √4 = 2.
Can a square root be negative?
A number has a negative square root as well as a positive one: 7 × 7 = 49 and (−7) × (−7) = 49. The √ sign stands for the positive root only, so √49 = 7 and the negative root is written −√49 = −7. A negative number such as −4 has no real square root.
