√448 at a glance
- Exact value
- 8√7
- Decimal (10 places)
- 21.1660104885
- Rounded
- 21.2 · 21.17 · 21.166
- Perfect square?
- No — between 21² and 22²
- Rational?
- Irrational
- Both square roots
- ±21.166010
- Prime factorization
- 2⁶ × 7
- Cube root
- 7.651725
How to simplify √448
Look for the largest perfect square that divides 448. Here it is 64 (8²), because 448 = 64 × 7 and 7 has no square factor left:
The prime factorization tells the same story: 448 = 2⁶ × 7. Each pair of equal primes leaves the radical as one factor, so 2³ comes out and 7 stays inside.
448 has 3 square factors (4, 16 and 64). Starting with a smaller one still works but takes more rounds: √448 = 2√112, and √112 can be simplified again. Using 64 straight away finishes in one step.
Check: (8√7)² = 8² × 7 = 64 × 7 = 448. As a decimal, 8√7 = 8 × 2.6457513111 ≈ 21.1660104885.
Where √448 sits between perfect squares
441 = 21² and 484 = 22² are the nearest perfect squares, so √448 lies between 21 and 22. 448 is 7 above 441 and 36 below 484, so the root is closer to 21.
- Straight line between 441 and 484: 21.1628 (0.02% low)
- Tangent from 21, i.e. 21 + 7 ÷ 42: 21.1667 (0% high)
- Tangent from 22, i.e. 22 − 36 ÷ 44: 21.1818 (0.07% high)
For √448 the tangent at 21 wins, missing by only 0.0007. Tangent estimates shine when the number sits close to a perfect square — here 448 is just 7 above 441.
Finding √448 with the Babylonian method
Picture a rectangle with an area of 448 and one side x; the other side must be 448 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √448.
Start from the nearest whole number, 21 (21² = 441):
| Step | Guess x | 448 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 21.0000000000 | 21.3333333333 | 21.1666666667 | 3 |
| 2 | 21.1666666667 | 21.1653543307 | 21.1660104987 | 7 |
| 3 | 21.1660104987 | 21.1660104783 | 21.1660104885 | 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 √448 = 21.1660104885 to every decimal shown.
√448 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √448 the pattern is [21; 6, 42] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √448 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 |
|---|---|---|
| 21/1 | 21.0000000000 | 1.7 × 10⁻¹ |
| 127/6 | 21.1666666667 | 6.6 × 10⁻⁴ |
| 5,355/253 | 21.1660079051 | 2.6 × 10⁻⁶ |
| 32,257/1,524 | 21.1660104987 | 1.0 × 10⁻⁸ |
| 1,360,149/64,261 | 21.1660104885 | < 10⁻¹⁰ |
| 8,193,151/387,090 | 21.1660104885 | < 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 448y² = 1. Its smallest solution in positive whole numbers is x = 127, y = 6.
√448 in geometry and everyday measurements
- A square garage floor of 448 square feet measures about 21.17 ft (21 ft 2 in) per side, and its corner-to-corner diagonal is √896 ≈ 29.9 ft.
- 448 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 √448 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 √448 as its space diagonal.
- Since √448 = 8√7, a length of √448 is exactly 8 copies of the length √7 laid end to end.
Square roots near √448 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √445 | √445 | 21.0950 | No |
| √446 | √446 | 21.1187 | No |
| √447 | √447 | 21.1424 | No |
| √448 | 8√7 | 21.1660 | No |
| √449 | √449 | 21.1896 | No |
| √450 | 15√2 | 21.2132 | No |
| √451 | √451 | 21.2368 | No |
- The cube root of 448 is about 7.651725.
- Because 448 = 4 × 112, the root is twice √112: 2 × 10.583005 ≈ 21.16601.
Frequently asked questions
What is the square root of 448?
The square root of 448 is 8√7 in simplest radical form, which is about 21.1660104885. The negative root, −21.166010, also squares to 448.
Is the square root of 448 rational or irrational?
Irrational. 448 is not a perfect square — it falls between 441 and 484 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √448 be simplified?
Yes. The largest perfect square dividing 448 is 64, so √448 = √64 × √7 = 8√7.
What is √448 rounded to two decimal places?
√448 ≈ 21.17 to two decimal places (21.2 to one, 21.166 to three). Check: 21.17² = 448.1689, close to 448.