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Math & School
Square Root Calculator — Decimal Value & Simplified Radical Form
The decimal value, the simplified radical, and why negative numbers have no real square root
A square root of a number is the value that, multiplied by itself, gives that number back. Because 5 × 5 = 25, the square root of 25 is 5. The radical symbol √ stands for the principal (non-negative) square root, so √25 = 5, not −5 — even though both 5 and −5 square to 25. This calculator returns that principal root as an exact-as-possible decimal and, for square roots, the simplified radical form that your algebra class actually wants.
Perfect squares are the easy ones: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144… Their roots are whole numbers (√144 = 12). Every other positive number has an irrational square root — a decimal that never ends or repeats, like √2 = 1.41421356… or √10 = 3.16227766…. The decimal you see here is rounded, but the radical form is exact.
Simplifying a radical means pulling the largest perfect-square factor out from under the root. The rule is √(a·b) = √a · √b, so you factor the number, take out anything that's a perfect square, and leave the rest inside. For example √72: the largest perfect square that divides 72 is 36, and 72 = 36 × 2, so √72 = √36 · √2 = 6√2. Likewise √50 = √(25·2) = 5√2, √48 = √(16·3) = 4√3, and √200 = √(100·2) = 10√2. If no perfect square (other than 1) divides the number — as with √7, √13 or √15 — the radical is already in simplest form. This calculator shows that step so you can check your homework, not just copy an answer.
The calculator also handles other roots. Switch the index to get a cube root (∛, the number that cubed gives your value — ∛27 = 3) or any nth root at all. Unlike square roots, odd roots like the cube root can take negative numbers: ∛(−8) = −2, because (−2)³ = −8.
One hard rule for square roots and all even roots: a negative number has no real square root. There's no real value whose square is −9, because squaring any real number gives a non-negative result. In that case the answer lives in the complex numbers (√−9 = 3i), which this calculator flags rather than computes. Enter a non-negative number for square roots, and use an odd index if you need the root of a negative.
📖 See also: The Quadratic Formula, Explained Step by Step
Calculator
Fill in the fields and click "Calculate" for instant results.
📰 Formula
• Square root: √n = the non-negative x with x² = n (n ≥ 0) • nth root: x = n^(1/index), the value with x^index = n • Simplify a radical: √n = √(p² · k) = p · √k, where p² is the largest perfect-square factor of n • Product rule: √(a · b) = √a · √b • Negatives: √(−n) has no real value; √(−n) = √n · i (imaginary). Odd roots allow negatives: ∛(−8) = −2
📰 Formula
• Square root: √n = the non-negative x with x² = n (n ≥ 0) • nth root: x = n^(1/index), the value with x^index = n • Simplify a radical: √n = √(p² · k) = p · √k, where p² is the largest perfect-square factor of n • Product rule: √(a · b) = √a · √b • Negatives: √(−n) has no real value; √(−n) = √n · i (imaginary). Odd roots allow negatives: ∛(−8) = −2
🧪 Worked examples
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Example 2
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Example 3
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Example 4
⚠️ Common mistakes
- Writing the negative root too: √25 = 5 only; −5 is the other solution to x² = 25, but √ means the principal (non-negative) root.
- Stopping at a small factor instead of the largest perfect square — √72 = 2√18 is incomplete; finish it to 6√2.
- Trying to take the square root of a negative number and expecting a real answer (it's imaginary).
- Confusing the square root with halving — √36 is 6, not 18.
- Forgetting that the cube root of a negative is real: ∛(−27) = −3, not 'no solution'.
💡 Tips
- To simplify by hand, find the biggest perfect square (4, 9, 16, 25, 36, 49, 64…) that divides your number, then take its root outside the radical.
- If no perfect square other than 1 divides the number, the radical is already simplest — √7, √13 and √15 can't be reduced.
- Estimate first: √50 is just above √49 = 7, so expect roughly 7.07 — a quick sanity check on the decimal.
- Square roots and division by even roots need a non-negative input; use an odd index (3, 5, 7…) when you must root a negative number.
Embed this calculator on your site
Copy the code below and paste it into the HTML of your site or blog.
<iframe src="https://www.calcnimbus.com/embed/square-root-calculator" width="100%" height="500" frameborder="0" style="border:1px solid #eee;border-radius:12px"></iframe>
❓ Frequently asked questions
What is the square root of a number?
It's the value that, multiplied by itself, equals that number. Since 7 × 7 = 49, the square root of 49 is 7. The √ symbol means the non-negative (principal) root.
How do I write a square root in simplified radical form?
Factor out the largest perfect square. For √72, the biggest perfect square dividing 72 is 36, and 72 = 36 × 2, so √72 = √36 · √2 = 6√2. If no perfect square divides the number, it's already simplest.
Can you take the square root of a negative number?
Not within the real numbers — no real value squared gives a negative. The result is imaginary: √(−9) = 3i. This calculator flags negative inputs for square roots instead of returning a fake real answer.
Is the square root of 25 equal to 5 or both 5 and −5?
The principal square root √25 is 5. Both 5 and −5 are solutions to the equation x² = 25, but the radical symbol by itself refers only to the non-negative root.
What's the difference between a square root and a cube root?
A square root undoes squaring (x² = n), while a cube root undoes cubing (x³ = n). Cube roots accept negatives because an odd power of a negative is negative: ∛(−8) = −2. Switch the index field to compute cube and higher roots.
Why is the square root of 2 irrational?
There's no fraction or terminating decimal that squares exactly to 2, so √2 = 1.41421356… runs forever without repeating. The decimal shown here is rounded; the exact value is just '√2'.
How do I know if a number is a perfect square?
Its square root is a whole number with no remainder. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121 and 144 are perfect squares. If the root the calculator returns is an integer, the number is a perfect square.
How can I estimate a square root without a calculator?
Find the two nearest perfect squares. √50 sits just above √49 = 7, so it's about 7.07. √30 is between √25 = 5 and √36 = 6, closer to 5.5. This gives a quick mental ballpark.
How do I simplify √48 or √200?
√48 = √(16 × 3) = 4√3, because 16 is the largest perfect square dividing 48. √200 = √(100 × 2) = 10√2. Always pull out the biggest perfect-square factor to finish in one step.