Square Root Calculator
Enter any whole number below to instantly get its decimal value and simplified radical form.
Where √72 sits
8² = 64 and 9² = 81, and 72 falls between them, so √72 is between 8 and 9. It's closer to 8 (8 above 64, 9 below 81), which matches the decimal 8.485281.
| k | k² | Compared to 72 |
|---|---|---|
| 6 | 36 | 36 below |
| 7 | 49 | 23 below |
| 8 | 64 | 8 below |
| 9 | 81 | 9 above |
| 10 | 100 | 28 above |
| 11 | 121 | 49 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 8), divide 72 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 72 ÷ x | New guess (x + 72÷x) ÷ 2 |
|---|---|---|---|
| 1 | 8 | 9 | 8.5 |
| 2 | 8.5 | 8.470588 | 8.485294 |
| 3 | 8.485294 | 8.485269 | 8.485281 |
| 4 | 8.485281 | 8.485281 | 8.485281 |
After 4 passes the guess is 8.485281, already matching √72 = 8.485281 to six decimal places.
Exact form from the prime factors
72 = 2^3 × 3^2. Each pair of matching primes comes out of the root as one factor, and what's left stays inside: √72 = 6√2. That's the exact value; the decimal 8.485281 is a rounding of it.
What Is a Square Root?
The square root of a number is the value that, when multiplied by itself, produces that number. If the number is a perfect square (1, 4, 9, 16, 25...), its square root is a whole number. Otherwise, the square root is irrational — its decimal form never terminates or repeats — and it's usually expressed in simplified radical form instead.
How to Simplify a Square Root, Step by Step
- Find the prime factorization of the number.
- Group the prime factors into pairs of identical primes.
- For each pair, move one copy of that prime outside the radical sign.
- Multiply everything left outside together, and everything left inside together.
Worked example
Simplify √72.
Prime factorization: 72 = 2^3 × 3^2
The pair of 3s comes out as a single 3. Of the three 2s, one pair comes out as a single 2, leaving one 2 behind inside the radical.
√72 = 3 × 2 × √2 = 6√2 (≈ 8.485)
Square Roots of Popular Numbers
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Keep going
- For the roots you look up most, the table has every square root from 1 to 100 with its simplified radical form. Square roots table
- Square root and cube root use the same radical idea at different degrees; the cube root calculator solves the same kind of problem one degree higher. Cube Root Calculator
- Square root and absolute value are linked by definition -- √(x²) always equals |x|; the absolute value calculator shows that simpler side of the relationship. Absolute Value Calculator
Frequently Asked Questions
What is a square root?
The square root of a number is the value that, multiplied by itself, gives that number. The square root of 9 is 3, because 3 × 3 = 9. Every positive number has two square roots (one positive, one negative), but "the square root" usually refers to the positive one, called the principal square root.
Can a square root be negative?
Technically every positive number has two square roots — a positive one and a negative one (both 3 and −3 square to 9). By convention, the √ symbol always refers to the principal (positive) square root, so √9 = 3, not −3. You only see the negative root when solving an equation like x² = 9, which has two solutions: x = 3 and x = −3.
How do you simplify a square root by hand?
Find the prime factorization of the number. For every pair of identical prime factors, move one copy outside the radical. Whatever's left without a pair stays inside. For example, 72 = 2³ × 3² — the pair of 3s comes out as 3, and one 2 pairs up leaving one 2 inside: 72 = 2² × 2 × 3² → 6√2.
What is a perfect square?
A perfect square is a number that's the square of a whole number — 1, 4, 9, 16, 25, 36... Its square root is always a whole number with no radical needed.
Is the square root of a prime number always irrational?
Yes. A prime number's only factors are 1 and itself, so it can never be split into a pair of identical factors — its square root can't simplify to a whole number or a smaller radical, and it never terminates or repeats as a decimal.
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