Formula and how it works
ⁿ√x = x^(1/n) because (x^(1/n))ⁿ = x
The square root of x is the number that, multiplied by itself, gives x. Every positive number has two square roots (for 9: 3 and −3); the calculator shows the positive, principal root.
Simplifying a radical: split the number into a perfect power times a remainder. 72 = 36 × 2, and √36 = 6, so √72 = 6√2.
Worked examples
√144
12 × 12 = 144, so √144 = 12 – a perfect square.
Cube root of −27
(−3) × (−3) × (−3) = −27, so ∛−27 = −3. Odd roots of negative numbers are real.
Assumptions and limitations
- Even roots of negative numbers are not real numbers; the calculator shows the principal imaginary root (for example √−16 = 4i).
- Simplified radical forms are shown for whole numbers up to 1,000,000,000.
Frequently asked questions
How can I estimate a square root without a calculator?
Find the nearest perfect squares. √50 lies between √49 = 7 and √64 = 8, much closer to 7, so it is about 7.07.
Is the square root of 2 a fraction?
No. √2 ≈ 1.41421356 is irrational: its decimals never end or repeat, so it cannot be written exactly as a fraction.
What is the square root of 0?
0, because 0 × 0 = 0.
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