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Find and simplify square roots
Use the free square root calculator to work through the number relationship, preserve exact values, and check the result with estimation. Calculate with natural scientific notation, reusable history, angle modes, memory, probability functions, and familiar calculator layouts.
Arithmetic
Square Root Calculator explained
The short version
- The square root of a number is what you multiply by itself to get that number, so √49 = 7.
- Pull out the biggest perfect square hiding inside, and the answer stays exact: √72 = 6√2.
- The √ sign means only the positive root, but the equation x² = 49 has two answers, 7 and −7.
The formula this page uses
√a = the number that multiplies by itself to give a √(m² × n) = m√n if x² = a then x = ±√a
What each part means
SymbolWhat it means
a — The number under the signCalled the radicand. It cannot be negative if you want a real answer.
√ — The radical signAsks for the positive square root only, which is why √49 is 7 and not −7.
m² — The perfect square you pull outThe largest square factor. Getting the largest one first saves a second round.
± — Plus or minusWritten when you solve an equation by square-rooting both sides, because both signs work.
Show your work: a full example
- Simplify √72√72
- Find the biggest perfect square that divides 7272 = 36 × 2, and 36 is a perfect square
- Split the root across the multiplication√72 = √36 × √2
- Take the root of the square part√36 = 6, so √72 = 6√2
- Turn it into a decimal√2 = 1.4142, so 6 × 1.4142 = 8.4853
- Check by squaring8.4853 × 8.4853 = 72.0003, which is 72 once you allow for rounding
A second, different case
- A different case: two roots, and estimating by handx² = 49
- Both signs work7 × 7 = 49 and (−7) × (−7) = 49, so x = 7 or x = −7
- But the symbol itself picks one√49 = 7 only
- Now estimate √50 without a calculator7² = 49 and 8² = 64, so it is just over 7
- Try 7.17.1 × 7.1 = 50.41, slightly too big
- Try 7.077.07 × 7.07 = 49.9849, slightly too small
- Settle it√50 = 7.0711 to four decimal places, and in exact form 5√2
Copy-ready example
√72
Simplify √72
Questions about the Square Root Calculator
Why does √49 give one answer but x² = 49 give two?
Because they ask different questions. The √ symbol is defined to hand back the positive root only, so that it behaves like a proper function. The equation x² = 49 asks which numbers square to 49, and there are two of them.
Can I take the square root of a negative number?
Not within ordinary numbers, because any number times itself comes out positive or zero. Higher courses invent i, defined so that i² = −1, and then √(−9) = 3i. Until then, a negative under the root sign means the question has no real answer.
How do I estimate a square root by hand?
Trap it between two perfect squares first. For √50, since 49 and 64 sit either side, the answer is between 7 and 8 and much closer to 7. Then test one decimal at a time: 7.1 squared overshoots, so the answer sits between 7.0 and 7.1.
When should I leave the answer as a surd?
Whenever the question says exact, or when the root feeds into more working. Keeping 6√2 lets you square it later and land back on 72 exactly, while 8.4853 would drift. Convert to a decimal only for a final measurement someone has to use.
How to use a scientific calculator accurately
The calculator executes the expression entered; mathematical judgment still determines the correct expression, mode, and interpretation.
Set the angle mode
Use degrees for degree-labelled angles and radians for calculus or unit-circle work unless the problem states otherwise. A correct key sequence in the wrong mode produces a wrong value.
Group the expression
Use parentheses around numerators, denominators, exponents, and negative substitutions. Read the full display before evaluating.
Keep precision until the end
Reuse the stored answer instead of a rounded display value during multi-step work, then round once to the requested precision and units.
Estimate and verify
Predict sign and scale, use identities or inverse operations, and compare a second entry when the result matters.
About the calculator layouts
The MathGPT, TI-84 Plus-style, HP Prime-style, Casio fx-CG50-style, and Casio fx-9860GIII-style key arrangements share one independent calculation engine. They provide familiar practice layouts and do not emulate proprietary firmware or imply affiliation with those manufacturers.
Learn the mathematics
Understand arithmetic behind this calculator
A calculator confirms an answer. Working the method yourself is what makes the next problem faster.
GuidesLearn arithmeticConcepts, worked methods, and checks.
PracticePractice arithmetic questionsAnswer review after every attempt.
ReferenceArithmetic formulasConditions and examples for each formula.