Square Root Calculator

Enter one number, click Calculate. This square root calculator gives the root instantly, right on the page.

• Number (x) — the value you want the root of, such as 144 in the square root calculator's only box

With the default you get three results: the principal square root of 144 is exactly 12; the rounded root to the nearest whole number is 12; and the perfect-square check reads true, because $12^2=144$ exactly. Swap in any other number and the three rows update together — a decimal in the first row means the number is not a perfect square.

This page is more than a static square root calculator — it is also an AI tutor you can ask questions. The question box under the form takes full questions, photos of the problem, and follow-ups:

• Ask in plain words. “Simplify the square root of 72” or “Is 200 a perfect square?” The AI reads it and solves it step by step.
• Take a photo of your homework. No typing: the AI reads the numbers and solves it step by step.
• Ask about the ideas, not just the number. “Why does $\sqrt{20}$ simplify to $2\sqrt{5}$?” or “Why is there no real square root of a negative number?” The AI explains step by step — you can check every number against the square root calculator on this page, so nothing is guessed.
• Work backwards, then keep asking. Square an answer to test it, or ask which number squares to 121 — this square root calculator covers both directions. Follow-up questions are welcome, so ask “why?” as often as you need: the square root calculator page keeps teaching.

Squaring a number multiplies it by itself. A square root asks the reverse question: which number, times itself, gives this one?

$\sqrt{9}=3 \quad\text{because}\quad 3\times3=9$

Every positive number has two such numbers. Both $3$ and $-3$ square to 9, so both are roots of 9. By worldwide convention, the symbol $\sqrt{\ }$ and every square root calculator report the positive one — the principal square root. The other root is simply its negative.

That convention matters the moment you leave the calculator and solve equations. $\sqrt{9}=3$, always and only 3 — but the equation $x^2=9$ has two solutions, $x=3$ and $x=-3$. The radical sign returns one root; the equation demands both. Keeping those two habits separate is the most useful skill in this whole topic: when you see $\sqrt{\ }$, expect a single positive answer, and when you see $x^2=\text{something}$, expect two. The square root calculator above always shows the principal (positive) root in its first row.

A perfect square is a number whose square root is a whole number — the product of an integer times itself. These are the numbers whose roots come out exact, and they are worth knowing by heart:

Read the table in either direction. Left to right is squaring; right to left is taking roots: $\sqrt{144}=12$, $\sqrt{169}=13$, $\sqrt{225}=15$ — all exact, no decimals.

Type 144 into the square root calculator and the perfect-square row reads true; type 150 and it reads false, with a decimal root near 12. Two handy patterns: perfect squares can only end in 0, 1, 4, 5, 6, or 9 — never 2, 3, 7, or 8 — and the gaps between them grow ($13^2-12^2=25$), so they thin out as you climb. Both tricks help you guess where an unfamiliar root lands before computing it.

Roots come in two flavors. If the number is a perfect square, its root is exact: $\sqrt{144}=12$, full stop. Otherwise the root is irrational — a decimal that never ends and never repeats:

$\sqrt{2}\approx1.414213,\qquad \sqrt{20}\approx4.472136$

No finite decimal ever nails those values, and you cannot write $\sqrt{2}$ as a fraction — mathematicians proved that over two thousand years ago, and it remains the classic first example of an irrational number (this site has a full page on irrational numbers if you want the story). The result that shocked the ancient Greeks: a square with side 1 has a diagonal of exactly $\sqrt{2}$, a length that exists in geometry yet refuses to be written as a ratio.

So read a decimal root as an approximation, not the answer. $\sqrt{20}\approx4.472136$ is a rounding; the exact value is still available, because 20 hides a perfect-square factor. Pulling that factor out is called simplifying a radical, it is the next block's whole job, and the square root calculator's decimal row is how you check the result.

An irrational root can still be written exactly — in a shorter form. The trick: pull the largest perfect-square factor out of the radicand (the number under the root).

$\sqrt{20}=\sqrt{4\times5}=\sqrt{4}\times\sqrt{5}=2\sqrt{5}$

This works because a square root splits across multiplication — $\sqrt{a\times b}=\sqrt{a}\times\sqrt{b}$ — and the root of a perfect square is a whole number that can move out front. The method, step by step:

1. Factor the radicand and spot the largest perfect square dividing it. For 20 that is 4, from $20=4\times5$.
2. Split the root: $\sqrt{20}=\sqrt{4}\times\sqrt{5}$.
3. Evaluate the exact part: $\sqrt{4}=2$.
4. Write the result: $2\sqrt{5}$.

Insist on the largest perfect-square factor. A smaller one leaves work behind: $72=4\times18$ gives $2\sqrt{18}$, which still hides a perfect square, while $72=36\times2$ gives $6\sqrt{2}$, fully simplified in one pass. Check any result by squaring it: $(2\sqrt{5})^2=4\times5=20$. The square root calculator above confirms the decimal — $2\sqrt{5}$ and $\sqrt{20}$ both round to 4.472136.

Two routines cover most homework: verify an exact root, and simplify an irrational one.

Check that $\sqrt{144}=12$:

1. Square the proposed root: $12^2=144$.
2. It matches the radicand exactly, so 12 is the root — the square root calculator's perfect-square row reads true for 144.

Simplify $\sqrt{50}$:

1. Find the largest perfect-square factor of 50: since $50=25\times2$, it is 25.
2. Split the root: $\sqrt{50}=\sqrt{25}\times\sqrt{2}$.
3. Evaluate the exact part: $\sqrt{25}=5$.
4. Answer: $\sqrt{50}=5\sqrt{2}$.

Check the simplification the same way: $(5\sqrt{2})^2=25\times2=50$. Exact form and decimal form are one number in two outfits — the radical $5\sqrt{2}$ carries the value exactly, while the square root calculator's decimal row for 50 carries its rounding. When a test says “simplify”, give the radical; use the decimal to check yourself.

1. Splitting a root across a sum. $\sqrt{a+b}$ is not $\sqrt{a}+\sqrt{b}$. The classic counterexample: $\sqrt{9+16}=\sqrt{25}=5$, while $\sqrt{9}+\sqrt{16}=3+4=7$. Roots distribute over multiplication, never over addition.
2. Entering a negative number. No real number squares to $-9$, so $\sqrt{-9}$ has no real answer. The square root calculator reports that instead of inventing a fake decimal. (Negative radicands need complex numbers — a later course.)
3. Dropping the $\pm$ when solving equations. $\sqrt{9}=3$ alone, but $x^2=9$ has two solutions: $x=\pm3$. Forgetting the negative root loses half the answer.
4. Stopping at the decimal. $\sqrt{72}\approx8.485281$ is a rounding; $6\sqrt{2}$ is the exact value. When a problem says “simplify”, it wants the radical form — the square root calculator's decimal row is for checking, not for the final answer.