√703 at a glance
- Exact value
- √703
- Decimal (10 places)
- 26.5141471671
- Rounded
- 26.5 · 26.51 · 26.514
- Perfect square?
- No — between 26² and 27²
- Rational?
- Irrational
- Both square roots
- ±26.514147
- Prime factorization
- 19 × 37
- Cube root
- 8.891706
How to simplify √703
The prime factorization of 703 is 19 × 37. Every prime appears only once, so there is no pair to bring outside the radical — √703 is already in simplest form (a "square-free" radicand).
A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 703, 19 and 37 appear an odd number of times, so √703 is irrational and 26.5141471671 is a rounded value.
Where √703 sits between perfect squares
676 = 26² and 729 = 27² are the nearest perfect squares, so √703 lies between 26 and 27. 703 is 27 above 676 and 26 below 729, so the root is closer to 27.
- Straight line between 676 and 729: 26.5094 (0.02% low)
- Tangent from 26, i.e. 26 + 27 ÷ 52: 26.5192 (0.02% high)
- Tangent from 27, i.e. 27 − 26 ÷ 54: 26.5185 (0.02% high)
For √703 the tangent at 27 wins, missing by only 0.0044. Tangent estimates shine when the number sits close to a perfect square — here 703 is just 26 below 729.
Finding √703 with the Babylonian method
If a guess is too big, 703 ÷ 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√703) in one step.
Start from the nearest whole number, 27 (27² = 729):
| Step | Guess x | 703 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 27.0000000000 | 26.0370370370 | 26.5185185185 | 2 |
| 2 | 26.5185185185 | 26.5097765363 | 26.5141475274 | 6 |
| 3 | 26.5141475274 | 26.5141468068 | 26.5141471671 | all 10 shown |
The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √703 = 26.5141471671 to every decimal shown.
√703 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √703 the pattern is [26; 1, 1, 17, 5, 1, 5, 17, 1, 1, 52] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √703 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 |
|---|---|---|
| 26/1 | 26.0000000000 | 5.1 × 10⁻¹ |
| 27/1 | 27.0000000000 | 4.9 × 10⁻¹ |
| 53/2 | 26.5000000000 | 1.4 × 10⁻² |
| 928/35 | 26.5142857143 | 1.4 × 10⁻⁴ |
| 4,693/177 | 26.5141242938 | 2.3 × 10⁻⁵ |
| 5,621/212 | 26.5141509434 | 3.8 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 703y² = 1. Its smallest solution in positive whole numbers is x = 1,159,172, y = 43,719.
√703 in geometry and everyday measurements
- 703 square feet is 65.3 m². Laid out as a square — a small house footprint or a lot — it is about 26.51 ft (26 ft 6 in) on a side.
- 703 is not a sum of two whole-number squares — the prime factor 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √703 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 √703 as its space diagonal.
Square roots near √703 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √700 | 10√7 | 26.4575 | No |
| √701 | √701 | 26.4764 | No |
| √702 | 3√78 | 26.4953 | No |
| √703 | √703 | 26.5141 | No |
| √704 | 8√11 | 26.5330 | No |
| √705 | √705 | 26.5518 | No |
| √706 | √706 | 26.5707 | No |
- The cube root of 703 is about 8.891706.
- Squaring undoes the root: (√703)² = 703, while 703² = 494,209 — the number whose square root is 703.
Frequently asked questions
What is the square root of 703?
The square root of 703 is √703, about 26.5141471671. The negative root, −26.514147, also squares to 703.
Is the square root of 703 rational or irrational?
Irrational. 703 is not a perfect square — it falls between 676 and 729 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √703 be simplified?
No. 703 = 19 × 37 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √703 rounded to two decimal places?
√703 ≈ 26.51 to two decimal places (26.5 to one, 26.514 to three). Check: 26.51² = 702.7801, close to 703.