Math
Square root calculator
The decimal answer is easy. The exact one — 12 as 2 times root 3 — is what homework usually wants.
Exact form versus decimal
√72 is 8.485281... and it never terminates or repeats, because 72 is not a perfect square. That decimal is fine for measuring something, but it is an approximation, and school algebra usually wants the exact value instead.
The exact form comes from pulling out the largest perfect square factor. Since 72 = 36 × 2, and √36 = 6, we can write √72 = 6√2. That expression is precise — it loses nothing — and it is easier to work with in further algebra, because surds combine cleanly: 6√2 + 3√2 = 9√2, which is not obvious from the decimals.
How simplification works
Find the largest square that divides the number, take its root outside the radical, and leave the rest inside:
- √50 = √(25 × 2) = 5√2
- √48 = √(16 × 3) = 4√3
- √98 = √(49 × 2) = 7√2
- √30 stays as √30 — its factors are 2, 3 and 5, none repeated, so nothing comes out
A quicker route for larger numbers is prime factorisation. Break the number into primes, then every pair of identical primes contributes one copy outside the root. 72 = 2³ × 3² gives a pair of 2s and a pair of 3s, so 2 × 3 = 6 comes out and a single 2 stays in.
Negative numbers and even roots
No real number squared gives a negative result, since a negative times a negative is positive. So √−9 has no real value, and the calculator refuses it rather than returning NaN. In complex arithmetic it equals 3i, but that is a different number system.
Odd roots have no such problem. The cube root of −27 is −3, because (−3)³ = −27. This is why the calculator accepts negatives for cube and fifth roots but not for square and fourth.
Estimating in your head
Bracket the number between the perfect squares either side, which the calculator shows. For √72: 8² = 64 and 9² = 81, so the answer is between 8 and 9. Since 72 is roughly two-thirds of the way from 64 to 81, the root is a bit under 8.5 — and it is 8.485.
This estimate is worth doing before trusting any calculator, including this one. It catches the most common input error, which is a mistyped digit.
Common questions
How do I simplify a square root?
Find the largest perfect square that divides the number, take its root outside the radical, and leave the remainder inside. For 72, the largest square factor is 36, so the answer is 6 root 2.
Why can I not take the square root of a negative number?
Because any real number multiplied by itself is positive or zero. There is no real x with x squared equal to -9. In complex numbers the answer is 3i, but that lies outside real arithmetic.
Can I take the cube root of a negative number?
Yes. Odd roots preserve sign, so the cube root of -27 is -3. Only even roots — square, fourth, sixth — fail on negative inputs.
What is the difference between a square root and a radical?
The radical sign is the symbol itself. A square root is the specific case where the index is 2. The same symbol with a small 3 means cube root, and so on for higher indices.