Square Root Calculator
Find the square root (√n), cube root (∛n) or any nth root instantly. Returns simplified surd form, 10-decimal precision and rational/irrational classification.
Calculate Square Root
Square Root Table: 1 to 100
Click any row to load it into the calculator. Green = perfect square (rational).
| n | √n (decimal) | Simplified | Type |
|---|
What is a Square Root?
The square root of a number n is the value that, when multiplied by itself, gives n. It is written using the radical sign √ — also called the under root symbol. For example:
√16 = 4 because 4 × 4 = 16
√2 ≈ 1.4142135624 (irrational — never ends)
Every positive number has two square roots: a positive root (called the principal root) and a negative root. The √ symbol always refers to the positive (principal) square root. So √25 = 5, even though (−5)² = 25 too.
Zero has one square root: √0 = 0. Negative numbers have no real square root (they involve imaginary numbers, which is beyond GCSE scope).
How to Find a Square Root — Prime Factorisation Method
For perfect squares, prime factorisation gives the exact answer without a calculator. For non-perfect squares, you get a simplified surd (radical expression).
Step-by-step example: √72
- Prime factorise the radicand: 72 = 2 × 2 × 2 × 3 × 3
- Group into pairs: (2 × 2) × 2 × (3 × 3)
- Take one from each pair outside the √: 2 × 3 = 6
- Remaining factor stays inside: 2
- Result: √72 = 6√2 ≈ 8.4852813742
72 = 36 × 2
√72 = √36 × √2 = 6√2
6 × 1.4142135624 ≈ 8.4853
Example: √8
√8 = √4 × √2 = 2√2 ≈ 2.8284271247
Example: √27
√27 = √9 × √3 = 3√3 ≈ 5.1961524227
The rule used is the product rule for radicals: √(a × b) = √a × √b. This only works when both a and b are non-negative.
What are √2, √3, √5? — Irrational Numbers Explained
When a number is not a perfect square, its square root is irrational — meaning the decimal goes on forever without repeating. You cannot write it as a fraction p/q where p and q are integers.
- √2 = 1.41421356237309504880168872... (proven irrational by the ancient Greeks)
- √3 = 1.73205080756887729352744634...
- √5 = 2.23606797749978969640917366...
- √7 = 2.64575131106459059050161575...
- √10 = 3.16227766016837933199889354...
- √15 = 3.87298334620741688517927...
(15 = 3 × 5, no perfect-square factor, so √15 cannot be simplified)
Rational square roots — those with exact, terminating decimal values — only occur for perfect squares (1, 4, 9, 16, 25 …) and their fractions. Every other positive integer has an irrational square root.
Square Root of Perfect Square Numbers (1 to 225)
A perfect square is an integer that is the square of another integer. Its square root is always a whole number (rational). Here are all perfect squares up to 225:
Perfect squares are important in GCSE maths because they appear frequently in simplification, Pythagoras' theorem, quadratics, and area problems.
The sequence of perfect squares is: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400 ...
To check if a number is a perfect square, take its square root — if the result is a whole number, it is a perfect square.
Cube Root Explained — ∛n
The cube root of a number n is the value that, when multiplied by itself three times, gives n. It is written ∛n (or ³√n).
∛8 = 2 because 2 × 2 × 2 = 8
∛64 = 4 because 4 × 4 × 4 = 64
∛125 = 5 because 5 × 5 × 5 = 125
Unlike square roots, cube roots of negative numbers are real: ∛(−8) = −2 because (−2)³ = −8.
Common perfect cubes
| n | n³ | ∛(n³) |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 8 | 2 |
| 3 | 27 | 3 |
| 4 | 64 | 4 |
| 5 | 125 | 5 |
| 6 | 216 | 6 |
| 7 | 343 | 7 |
| 8 | 512 | 8 |
| 9 | 729 | 9 |
| 10 | 1000 | 10 |
The question "27 square root cubed" is often interpreted as either ∛27 = 3, or (√27)³ = (3√3)³ = 81√3 ≈ 140.3. Our calculator handles both via the nth root input.
How to Find a Square Root on a Calculator — the √ Button
The key sequence depends on your calculator type. Here are the most common methods for using sqrt on a calculator:
Basic (non-scientific) calculator
- Type the number (e.g. 64)
- Press √
- Read the result (8)
Scientific calculator (e.g. Casio fx-83GT, fx-85GT)
- Press √
- Type the number (e.g. 8)
- Press =
- Result: 2√2 or 2.828... (depending on mode)
On Casio calculators in MathIO mode, the display shows exact surd form (e.g. 2√2). Switch to LineIO mode for a decimal result.
Windows Calculator (PC)
- Open Calculator → Scientific mode
- Type the number
- Click √ or press @ on the keyboard
Keyboard shortcut
In Microsoft Excel or Google Sheets, use =SQRT(number). For example, =SQRT(8) returns 2.828427...
Square Root on a Phone Calculator — Android & iPhone (iOS)
iPhone (iOS)
- Open the Calculator app
- Rotate your phone to landscape (turn sideways) to reveal the scientific layout
- Enter the number, then tap √x
- Alternatively: type the number and tap the √ button
If your screen rotation is locked, swipe down from the top-right corner to open Control Centre and disable the rotation lock first.
Android
- Open the Calculator app (stock Google Calculator or Samsung Calculator)
- Tap the three-dot menu or swipe up/rotate to switch to Scientific mode
- Tap √, then enter the number, then tap =
If you cannot find the scientific mode, search for "Scientific Calculator" on the Google Play Store — there are many free options.
Square Root in GCSE Maths
Square roots appear throughout the GCSE Maths curriculum in England, Wales and Northern Ireland (AQA, Edexcel, OCR exam boards). Key topics include:
- Number — recognising perfect squares and their roots (1 to 225); listing integer values for inequalities involving n²
- Surds (Higher tier) — simplifying expressions such as √8 = 2√2; rationalising denominators; expanding brackets with surds
- Pythagoras' Theorem — c = √(a² + b²), for example a 3-4-5 right triangle: c = √(9+16) = √25 = 5
- Quadratic equations — using the quadratic formula, which involves √(b²−4ac)
- Area and volume — finding the side of a square from its area: side = √area
- Trigonometry — exact values: sin 45° = cos 45° = √2/2; tan 60° = √3
According to the BBC Bitesize GCSE Maths guide, students should be able to evaluate square roots of perfect squares without a calculator and simplify surds at Higher tier.
Exact GCSE trigonometric values involving square roots
| Angle | sin | cos | tan |
|---|---|---|---|
| 30° | 1/2 | √3/2 | 1/√3 = √3/3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
Surds Explained — Why √2 Cannot Be Written as a Fraction
A surd is any root that is irrational — one whose decimal expansion never terminates or repeats. √2, √3, √5 and √7 are all surds. Perfect-square roots (√4 = 2, √9 = 3) are not surds because they are rational.
Proof that √2 is irrational (simplified)
Assume √2 = p/q where p and q are integers with no common factor (the fraction is in lowest terms). Then 2 = p²/q², so p² = 2q². This means p² is even, so p must be even. Write p = 2k. Then (2k)² = 2q², giving 4k² = 2q², so q² = 2k², meaning q is also even. But if both p and q are even, they share a factor of 2 — contradicting our assumption. Therefore √2 cannot be written as p/q. It is irrational.
Key surd rules for GCSE
√(a / b) = √a / √b e.g. √(1/4) = 1/2
(√a)² = a e.g. (√5)² = 5
a√b + c√b = (a+c)√b e.g. 3√2 + 5√2 = 8√2
(√a + √b)(√a − √b) = a − b (difference of squares)
Surds you will encounter frequently
- 5√5 = √125 ≈ 11.1803 (appears in pentagon geometry)
- 8√8 = 16√2 ≈ 22.627 (simplify: 8√8 = 8 · 2√2 = 16√2)
- 3√3 = √27 ≈ 5.196 (appears in equilateral triangle height)
- 2√2 = √8 ≈ 2.828 (diagonal of a unit square = √2; diagonal of a 2×2 square = 2√2)
Frequently Asked Questions
What is the square root of a square number?
A square number (perfect square) has a whole-number square root. √1=1, √4=2, √9=3, √16=4, √25=5, √36=6, √49=7, √64=8, √81=9, √100=10. These are rational results — no decimal places needed.
What is √8 simplified?
√8 simplified is 2√2. Since 8 = 4 × 2, we get √8 = √4 × √2 = 2√2. As a decimal: √8 ≈ 2.8284271247.
How do you find √27?
√27 simplified is 3√3. Write 27 = 9 × 3: √27 = √9 × √3 = 3√3 ≈ 5.1961524227. Do not confuse this with the cube root of 27: ∛27 = exactly 3.
What is the square root of 15?
√15 ≈ 3.8729833462. It cannot be simplified because 15 = 3 × 5 has no perfect-square factor. √15 remains in surd form as √15.
What does √ mean on a calculator?
The √ button (or sqrt button) calculates the positive square root of the displayed number. Entering 64 then pressing √ gives 8. On scientific calculators (e.g. Casio), press √ first, then the number.
How do you type a square root symbol?
- Windows: Hold Alt and type 251 on the numeric keypad → √
- Mac: Press Option + V → √
- iPhone / Android: Long-press the letter v on the keyboard; √ appears in the pop-up
- HTML: Use
√or√
What is 5 square root 5 (5√5)?
5√5 means 5 multiplied by √5. Since √5 ≈ 2.2360679775, we get 5 × 2.2360679775 ≈ 11.1803398875. In surd form, 5√5 is the simplest form of √125 (because 125 = 25 × 5, so √125 = 5√5).
What is 8 square root 8 (8√8)?
8√8 = 8 × √8 = 8 × 2√2 = 16√2 ≈ 22.6274169979. So 8√8 simplifies neatly to 16√2.
What is under root 10?
"Under root 10" means √10 (10 under the root sign) ≈ 3.1622776602. It cannot be simplified and is an irrational number. In surd form it stays as √10.
What is the square root of square root (√√n)?
The square root of a square root is the fourth root: √(√n) = ⁴√n = n^(1/4). For example, √(√16) = √4 = 2 (which is the same as 16^0.25 = 2). Use the nth root input above — set n = 4 — to calculate fourth roots.