√32 at a glance
- Exact value
- 4√2
- Decimal (10 places)
- 5.6568542495
- Rounded
- 5.7 · 5.66 · 5.657
- Perfect square?
- No — between 5² and 6²
- Rational?
- Irrational
- Both square roots
- ±5.656854
- Prime factorization
- 2⁵
- Cube root
- 3.174802
How to simplify √32
Look for the largest perfect square that divides 32. Here it is 16 (4²), because 32 = 16 × 2 and 2 has no square factor left:
The prime factorization tells the same story: 32 = 2⁵. Each pair of equal primes leaves the radical as one factor, so 2² comes out and 2 stays inside.
32 has 2 square factors (4 and 16). Starting with a smaller one still works but takes more rounds: √32 = 2√8, and √8 can be simplified again. Using 16 straight away finishes in one step.
Check: (4√2)² = 4² × 2 = 16 × 2 = 32. As a decimal, 4√2 = 4 × 1.4142135624 ≈ 5.6568542495.
Where √32 sits between perfect squares
25 = 5² and 36 = 6² are the nearest perfect squares, so √32 lies between 5 and 6. 32 is 7 above 25 and 4 below 36, so the root is closer to 6.
- Straight line between 25 and 36: 5.6364 (0.36% low)
- Tangent from 5, i.e. 5 + 7 ÷ 10: 5.7000 (0.76% high)
- Tangent from 6, i.e. 6 − 4 ÷ 12: 5.6667 (0.17% high)
For √32 the tangent at 6 wins, missing by only 0.0098. Tangent estimates shine when the number sits close to a perfect square — here 32 is just 4 below 36.
Finding √32 with the Babylonian method
Picture a rectangle with an area of 32 and one side x; the other side must be 32 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √32.
Start from the nearest whole number, 6 (6² = 36):
| Step | Guess x | 32 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 6.0000000000 | 5.3333333333 | 5.6666666667 | 2 |
| 2 | 5.6666666667 | 5.6470588235 | 5.6568627451 | 5 |
| 3 | 5.6568627451 | 5.6568457539 | 5.6568542495 | all 10 shown |
The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √32 = 5.6568542495 to every decimal shown.
√32 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √32 the pattern is [5; 1, 1, 1, 10] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √32 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 |
|---|---|---|
| 5/1 | 5.0000000000 | 6.6 × 10⁻¹ |
| 6/1 | 6.0000000000 | 3.4 × 10⁻¹ |
| 11/2 | 5.5000000000 | 1.6 × 10⁻¹ |
| 17/3 | 5.6666666667 | 9.8 × 10⁻³ |
| 181/32 | 5.6562500000 | 6.0 × 10⁻⁴ |
| 198/35 | 5.6571428571 | 2.9 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 32y² = 1. Its smallest solution in positive whole numbers is x = 17, y = 3.
√32 in geometry and everyday measurements
- A square room or garden bed covering 32 square feet measures about 5.66 ft (5 ft 8 in) along each wall.
- 32 = 4² + 4², so by the Pythagorean theorem √32 is the diagonal of a 4 × 4 rectangle — and the distance between the points (0, 0) and (4, 4) on a grid.
- Since √32 = 4√2, a length of √32 is exactly 4 copies of the length √2 laid end to end.
Square roots near √32 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √29 | √29 | 5.3852 | No |
| √30 | √30 | 5.4772 | No |
| √31 | √31 | 5.5678 | No |
| √32 | 4√2 | 5.6569 | No |
| √33 | √33 | 5.7446 | No |
| √34 | √34 | 5.8310 | No |
| √35 | √35 | 5.9161 | No |
- The cube root of 32 is about 3.174802.
- Four times the radicand doubles the root: √128 = 2 × √32 ≈ 11.313708.
Frequently asked questions
What is the square root of 32?
The square root of 32 is 4√2 in simplest radical form, which is about 5.6568542495. The negative root, −5.656854, also squares to 32.
Is the square root of 32 rational or irrational?
Irrational. 32 is not a perfect square — it falls between 25 and 36 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √32 be simplified?
Yes. The largest perfect square dividing 32 is 16, so √32 = √16 × √2 = 4√2.
What is √32 rounded to two decimal places?
√32 ≈ 5.66 to two decimal places (5.7 to one, 5.657 to three). Check: 5.66² = 32.0356, close to 32.