Simplest radical form
√75 = √(25 × 3) = 5√3
Long division method
- Group the digits in pairs from the decimal point: 75. The largest square ≤ 75 is 8² = 64, so the first digit is 8; remainder 11.
- Bring down 00: 1100. Double the root so far (8 → 16) and find the largest d with (16d) × d ≤ 1100: d = 6, since 166 × 6 = 996. Remainder 104.
- Bring down 00: 10400. Double the root so far (86 → 172) and find the largest d with (172d) × d ≤ 10400: d = 6, since 1726 × 6 = 10356. Remainder 44.
- Bring down 00: 4400. Double the root so far (866 → 1732) and find the largest d with (1732d) × d ≤ 4400: d = 0, since 17320 × 0 = 0. Remainder 4400.
The digits read 8.660, so √75 ≈ 8.660.
Quick estimate
8² = 64 and 9² = 81, so √75 lies between 8 and 9. A fast first guess is 8 + 11/(2 × 8) = 8.688, against the true 8.660.
Square roots of other numbers
| Root | Value |
|---|---|
| √2 | 1.4142 |
| √3 | 1.7321 |
| √5 | 2.2361 |
| √6 | 2.4495 |
| √7 | 2.6458 |
| √8 | 2.8284 |
| √10 | 3.1623 |
| √11 | 3.3166 |
| √12 | 3.4641 |
| √13 | 3.6056 |
| √14 | 3.7417 |
| √15 | 3.8730 |
| √17 | 4.1231 |
| √18 | 4.2426 |
| √19 | 4.3589 |
| √20 | 4.4721 |
| √21 | 4.5826 |
| √22 | 4.6904 |
| √23 | 4.7958 |
| √24 | 4.8990 |
| √26 | 5.0990 |
| √27 | 5.1962 |
| √28 | 5.2915 |
| √29 | 5.3852 |
| √30 | 5.4772 |
| √31 | 5.5678 |
| √32 | 5.6569 |
| √33 | 5.7446 |
| √34 | 5.8310 |
| √35 | 5.9161 |
| √37 | 6.0828 |
| √38 | 6.1644 |
| √39 | 6.2450 |
| √40 | 6.3246 |
| √41 | 6.4031 |
| √42 | 6.4807 |
| √43 | 6.5574 |
| √44 | 6.6332 |
| √45 | 6.7082 |
| √46 | 6.7823 |
| √47 | 6.8557 |
| √48 | 6.9282 |
| √50 | 7.0711 |
| √72 | 8.4853 |
| √75 | 8.6603 |
| √80 | 8.9443 |
| √98 | 9.8995 |
| √125 | 11.1803 |
| √200 | 14.1421 |
Questions students ask
What is the square root of 75?
√75 ≈ 8.6603. It is irrational, so the decimal never ends or repeats.
Is √75 rational?
No. 75 is not a perfect square, so √75 is irrational.
What is √75 in simplest radical form?
√75 = √(25 × 3) = 5√3
How do you find √75 by long division?
Pair the digits, find the largest square under the first pair, then repeatedly double the root so far and find the next digit. It gives √75 ≈ 8.660.