Root Calculator
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√72 = 8.485281374 = 6√2
1. Prime factors: 72 = 2³ × 3² 2. Each group of 2 equal factors comes out as one factor: 2² → 2, 3² → 3 3. Outside the radical: 6; left inside: 2 4. So √72 = 6√2 ≈ 8.48528137424 5. Check: (8.48528137424)² = 72
| n | n² |
|---|---|
| 6 | 36 |
| 7 | 49 |
| 8 | 64 |
| 9 | 81 |
| 10 | 100 |
| 11 | 121 |
How to use the root calculator
- Choose square root, cube root, or n-th root and enter the index (4 for a fourth root, 5 for a fifth root, and so on).
- Enter the number under the radical sign (the radicand). Negative numbers are fine for odd roots.
- Read the root, its simplified radical form for whole numbers, and the factor-by-factor working. The table lists nearby perfect powers to check that the answer makes sense.
Root formulas
ⁿ√(a × b) = ⁿ√a × ⁿ√b · ⁿ√(aⁿ × b) = a × ⁿ√b
Even n: x ≥ 0, two real roots ±r · Odd n: one real root, same sign as x
The radicand x is the number under the sign and the index n says which root: 2 for square, 3 for cube. Taking the n-th root undoes raising to the n-th power. Even roots of negative numbers are not real, because any real number raised to an even power is positive or zero; the square root of a negative number is imaginary, √(−9) = 3i. Odd roots keep the sign: ∛(−125) = −5.
Worked example: simplifying √72
Factor 72 into primes: 72 = 2³ × 3². A square root takes one factor out for every pair, so 2² comes out as 2 and 3² comes out as 3, leaving a single 2 inside. That gives √72 = 2 × 3 × √2 = 6√2, which is about 8.485281374. The same idea works for other roots with groups of three or four: ∛54 = ∛(3³ × 2) = 3∛2, and ⁴√81 = 3 because 3⁴ = 81 (so −3 is a fourth root of 81 too).
Estimating a square root by hand
Heron's method (also called the Babylonian method) improves a guess g for √s by replacing it with the average of g and s ÷ g. Starting from 1 for √2: 1 → 1.5 → 1.41666666667 → 1.41421568627 → 1.41421356237. Each step roughly doubles the number of correct digits, so four steps from a rough guess already match √2 to 11 decimal places.
Square and cube roots of common numbers
| x | √x simplified | √x | ∛x simplified | ∛x |
|---|---|---|---|---|
| 2 | √2 | 1.4142136 | ∛2 | 1.259921 |
| 3 | √3 | 1.7320508 | ∛3 | 1.4422496 |
| 5 | √5 | 2.236068 | ∛5 | 1.7099759 |
| 8 | 2√2 | 2.8284271 | 2 | 2 |
| 12 | 2√3 | 3.4641016 | ∛12 | 2.2894285 |
| 18 | 3√2 | 4.2426407 | ∛18 | 2.6207414 |
| 20 | 2√5 | 4.472136 | ∛20 | 2.7144176 |
| 27 | 3√3 | 5.1961524 | 3 | 3 |
| 32 | 4√2 | 5.6568542 | 2∛4 | 3.1748021 |
| 48 | 4√3 | 6.9282032 | 2∛6 | 3.6342412 |
| 50 | 5√2 | 7.0710678 | ∛50 | 3.6840315 |
| 64 | 8 | 8 | 4 | 4 |
| 72 | 6√2 | 8.4852814 | 2∛9 | 4.1601676 |
| 100 | 10 | 10 | ∛100 | 4.6415888 |
Tips and common mistakes
- The √ sign means the positive root. √81 = 9, but the equation x² = 81 has two solutions, 9 and −9.
- Roots don't split over addition. √(9 + 16) = √25 = 5, not √9 + √16 = 7. They do split over multiplication: √(4 × 18) = 2√18 = 6√2.
- Check with a power. Raise the answer to the index: 8.48528² ≈ 72. Small differences are rounding.
- Simplified form is exact; the decimal is not. Keep 6√2 in algebra and switch to decimals only at the end.
- For roots written as fractional powers, such as 8^(2/3), use the exponent calculator. For a mix of operations, try the scientific calculator.
Frequently asked questions
How do I simplify a square root?
Factor the number and take one factor out for every pair. 72 = 2³ × 3², so √72 = 2 × 3 × √2 = 6√2. If no prime appears twice, as with √17, the root is already as simple as it gets.
Can you take the cube root of a negative number?
Yes. Odd roots of negative numbers are real and negative, because a negative number cubed is negative: ∛(−125) = −5 since (−5)³ = −125.
What is the square root of a negative number?
It is not a real number, because every real number squared is 0 or positive. In complex numbers it is imaginary: √(−9) = 3i and √(−72) = 6i√2, where i² = −1.
Why does the square root of 81 have two answers?
Both 9² and (−9)² equal 81, so x² = 81 has two solutions. The √ sign means the principal (positive) root, so √81 = 9; the calculator also shows ±9 for even roots.
What is an nth root as an exponent?
The nth root of x is x to the power 1/n. So ⁴√81 = 81^(1/4) = 3, and the square root of 2 is 2^(1/2) ≈ 1.414213562.