√63 at a glance
- Exact value
- 3√7
- Decimal (10 places)
- 7.9372539332
- Rounded
- 7.9 · 7.94 · 7.937
- Perfect square?
- No — between 7² and 8²
- Rational?
- Irrational
- Both square roots
- ±7.937254
- Prime factorization
- 3² × 7
- Cube root
- 3.979057
How to simplify √63
Look for the largest perfect square that divides 63. Here it is 9 (3²), because 63 = 9 × 7 and 7 has no square factor left:
The prime factorization tells the same story: 63 = 3² × 7. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 7 stays inside.
Check: (3√7)² = 3² × 7 = 9 × 7 = 63. As a decimal, 3√7 = 3 × 2.6457513111 ≈ 7.9372539332.
Where √63 sits between perfect squares
49 = 7² and 64 = 8² are the nearest perfect squares, so √63 lies between 7 and 8. 63 is 14 above 49 and 1 below 64, so the root is closer to 8.
- Straight line between 49 and 64: 7.9333 (0.05% low)
- Tangent from 7, i.e. 7 + 14 ÷ 14: 8.0000 (0.79% high)
- Tangent from 8, i.e. 8 − 1 ÷ 16: 7.9375 (0% high)
For √63 the tangent at 8 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 63 is just 1 below 64.
Finding √63 with the Babylonian method
If a guess is too big, 63 ÷ 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√63) in one step.
Start from the nearest whole number, 8 (8² = 64):
| Step | Guess x | 63 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 8.0000000000 | 7.8750000000 | 7.9375000000 | 3 |
| 2 | 7.9375000000 | 7.9370078740 | 7.9372539370 | 8 |
| 3 | 7.9372539370 | 7.9372539294 | 7.9372539332 | all 10 shown |
The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √63 = 7.9372539332 to every decimal shown.
√63 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √63 the pattern is [7; 1, 14] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √63 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 |
|---|---|---|
| 7/1 | 7.0000000000 | 9.4 × 10⁻¹ |
| 8/1 | 8.0000000000 | 6.3 × 10⁻² |
| 119/15 | 7.9333333333 | 3.9 × 10⁻³ |
| 127/16 | 7.9375000000 | 2.5 × 10⁻⁴ |
| 1,897/239 | 7.9372384937 | 1.5 × 10⁻⁵ |
| 2,024/255 | 7.9372549020 | 9.7 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 63y² = 1. Its smallest solution in positive whole numbers is x = 8, y = 1.
√63 in geometry and everyday measurements
- A square room or garden bed covering 63 square feet measures about 7.94 ft (7 ft 11 in) along each wall.
- 63 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √63 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 √63 as its space diagonal.
- Since √63 = 3√7, a length of √63 is exactly 3 copies of the length √7 laid end to end.
Square roots near √63 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √60 | 2√15 | 7.7460 | No |
| √61 | √61 | 7.8102 | No |
| √62 | √62 | 7.8740 | No |
| √63 | 3√7 | 7.9373 | No |
| √64 | 8 | 8.0000 | Yes |
| √65 | √65 | 8.0623 | No |
| √66 | √66 | 8.1240 | No |
- The cube root of 63 is about 3.979057.
- Four times the radicand doubles the root: √252 = 2 × √63 ≈ 15.874508.
Frequently asked questions
What is the square root of 63?
The square root of 63 is 3√7 in simplest radical form, which is about 7.9372539332. The negative root, −7.937254, also squares to 63.
Is the square root of 63 rational or irrational?
Irrational. 63 is not a perfect square — it falls between 49 and 64 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √63 be simplified?
Yes. The largest perfect square dividing 63 is 9, so √63 = √9 × √7 = 3√7.
What is √63 rounded to two decimal places?
√63 ≈ 7.94 to two decimal places (7.9 to one, 7.937 to three). Check: 7.94² = 63.0436, close to 63.