√975 at a glance
- Exact value
- 5√39
- Decimal (10 places)
- 31.2249899920
- Rounded
- 31.2 · 31.22 · 31.225
- Perfect square?
- No — between 31² and 32²
- Rational?
- Irrational
- Both square roots
- ±31.224990
- Prime factorization
- 3 × 5² × 13
- Cube root
- 9.915962
How to simplify √975
Look for the largest perfect square that divides 975. Here it is 25 (5²), because 975 = 25 × 39 and 39 has no square factor left:
The prime factorization tells the same story: 975 = 3 × 5² × 13. Each pair of equal primes leaves the radical as one factor, so 5 comes out and 3 × 13 stays inside.
Check: (5√39)² = 5² × 39 = 25 × 39 = 975. As a decimal, 5√39 = 5 × 6.2449979984 ≈ 31.2249899920.
Where √975 sits between perfect squares
961 = 31² and 1,024 = 32² are the nearest perfect squares, so √975 lies between 31 and 32. 975 is 14 above 961 and 49 below 1,024, so the root is closer to 31.
- Straight line between 961 and 1,024: 31.2222 (0.01% low)
- Tangent from 31, i.e. 31 + 14 ÷ 62: 31.2258 (0% high)
- Tangent from 32, i.e. 32 − 49 ÷ 64: 31.2344 (0.03% high)
For √975 the tangent at 31 wins, missing by only 0.0008. Tangent estimates shine when the number sits close to a perfect square — here 975 is just 14 above 961.
Finding √975 with the Babylonian method
If a guess is too big, 975 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√975) in one step.
Start from the nearest whole number, 31 (31² = 961):
| Step | Guess x | 975 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 31.0000000000 | 31.4516129032 | 31.2258064516 | 3 |
| 2 | 31.2258064516 | 31.2241735537 | 31.2249900027 | 7 |
| 3 | 31.2249900027 | 31.2249899813 | 31.2249899920 | all 10 shown |
The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √975 = 31.2249899920 to every decimal shown.
√975 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √975 the pattern is [31; 4, 2, 4, 62] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √975 is irrational.
Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:
| Fraction | Decimal | Error |
|---|---|---|
| 31/1 | 31.0000000000 | 2.2 × 10⁻¹ |
| 125/4 | 31.2500000000 | 2.5 × 10⁻² |
| 281/9 | 31.2222222222 | 2.8 × 10⁻³ |
| 1,249/40 | 31.2250000000 | 1.0 × 10⁻⁵ |
| 77,719/2,489 | 31.2249899558 | 3.6 × 10⁻⁸ |
| 312,125/9,996 | 31.2249899960 | 4.0 × 10⁻⁹ |
The same fractions solve Pell’s equation, x² − 975y² = 1. Its smallest solution in positive whole numbers is x = 1,249, y = 40.
√975 in geometry and everyday measurements
- 975 square feet is 90.6 m². Laid out as a square — a small house footprint or a lot — it is about 31.22 ft (31 ft 3 in) on a side.
- 975 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √975 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √975 as its space diagonal.
- Since √975 = 5√39, a length of √975 is exactly 5 copies of the length √39 laid end to end.
Square roots near √975 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √972 | 18√3 | 31.1769 | No |
| √973 | √973 | 31.1929 | No |
| √974 | √974 | 31.2090 | No |
| √975 | 5√39 | 31.2250 | No |
| √976 | 4√61 | 31.2410 | No |
| √977 | √977 | 31.2570 | No |
| √978 | √978 | 31.2730 | No |
- The cube root of 975 is about 9.915962.
- Squaring undoes the root: (√975)² = 975, while 975² = 950,625 — the number whose square root is 975.
Frequently asked questions
What is the square root of 975?
The square root of 975 is 5√39 in simplest radical form, which is about 31.2249899920. The negative root, −31.224990, also squares to 975.
Is the square root of 975 rational or irrational?
Irrational. 975 is not a perfect square — it falls between 961 and 1024 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √975 be simplified?
Yes. The largest perfect square dividing 975 is 25, so √975 = √25 × √39 = 5√39.
What is √975 rounded to two decimal places?
√975 ≈ 31.22 to two decimal places (31.2 to one, 31.225 to three). Check: 31.22² = 974.6884, close to 975.