√45 at a glance
- Exact value
- 3√5
- Decimal (10 places)
- 6.7082039325
- Rounded
- 6.7 · 6.71 · 6.708
- Perfect square?
- No — between 6² and 7²
- Rational?
- Irrational
- Both square roots
- ±6.708204
- Prime factorization
- 3² × 5
- Cube root
- 3.556893
How to simplify √45
Look for the largest perfect square that divides 45. Here it is 9 (3²), because 45 = 9 × 5 and 5 has no square factor left:
The prime factorization tells the same story: 45 = 3² × 5. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 5 stays inside.
Check: (3√5)² = 3² × 5 = 9 × 5 = 45. As a decimal, 3√5 = 3 × 2.2360679775 ≈ 6.7082039325.
Where √45 sits between perfect squares
36 = 6² and 49 = 7² are the nearest perfect squares, so √45 lies between 6 and 7. 45 is 9 above 36 and 4 below 49, so the root is closer to 7.
- Straight line between 36 and 49: 6.6923 (0.24% low)
- Tangent from 6, i.e. 6 + 9 ÷ 12: 6.7500 (0.62% high)
- Tangent from 7, i.e. 7 − 4 ÷ 14: 6.7143 (0.09% high)
For √45 the tangent at 7 wins, missing by only 0.0061. Tangent estimates shine when the number sits close to a perfect square — here 45 is just 4 below 49.
Finding √45 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 45: following the tangent line down to zero simplifies to averaging x with 45 ÷ x.
Start from the nearest whole number, 7 (7² = 49):
| Step | Guess x | 45 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 7.0000000000 | 6.4285714286 | 6.7142857143 | 2 |
| 2 | 6.7142857143 | 6.7021276596 | 6.7082066869 | 5 |
| 3 | 6.7082066869 | 6.7082011781 | 6.7082039325 | 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 √45 = 6.7082039325 to every decimal shown.
√45 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √45 the pattern is [6; 1, 2, 2, 2, 1, 12] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √45 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 |
|---|---|---|
| 6/1 | 6.0000000000 | 7.1 × 10⁻¹ |
| 7/1 | 7.0000000000 | 2.9 × 10⁻¹ |
| 20/3 | 6.6666666667 | 4.2 × 10⁻² |
| 47/7 | 6.7142857143 | 6.1 × 10⁻³ |
| 114/17 | 6.7058823529 | 2.3 × 10⁻³ |
| 161/24 | 6.7083333333 | 1.3 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 45y² = 1. Its smallest solution in positive whole numbers is x = 161, y = 24.
√45 in geometry and everyday measurements
- A square room or garden bed covering 45 square feet measures about 6.71 ft (6 ft 8 in) along each wall.
- 45 = 3² + 6², so by the Pythagorean theorem √45 is the diagonal of a 3 × 6 rectangle — and the distance between the points (0, 0) and (3, 6) on a grid.
- Since √45 = 3√5, a length of √45 is exactly 3 copies of the length √5 laid end to end.
Square roots near √45 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √42 | √42 | 6.4807 | No |
| √43 | √43 | 6.5574 | No |
| √44 | 2√11 | 6.6332 | No |
| √45 | 3√5 | 6.7082 | No |
| √46 | √46 | 6.7823 | No |
| √47 | √47 | 6.8557 | No |
| √48 | 4√3 | 6.9282 | No |
- The cube root of 45 is about 3.556893.
- Four times the radicand doubles the root: √180 = 2 × √45 ≈ 13.416408.
Frequently asked questions
What is the square root of 45?
The square root of 45 is 3√5 in simplest radical form, which is about 6.7082039325. The negative root, −6.708204, also squares to 45.
Is the square root of 45 rational or irrational?
Irrational. 45 is not a perfect square — it falls between 36 and 49 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √45 be simplified?
Yes. The largest perfect square dividing 45 is 9, so √45 = √9 × √5 = 3√5.
What is √45 rounded to two decimal places?
√45 ≈ 6.71 to two decimal places (6.7 to one, 6.708 to three). Check: 6.71² = 45.0241, close to 45.