Square Root of 45
The square root of 45 is approximately 6.708204, or exactly 3√5 in simplified radical form. No, 45 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √45 sits
6² = 36 and 7² = 49, and 45 falls between them, so √45 is between 6 and 7. It's closer to 7 (9 above 36, 4 below 49), which matches the decimal 6.708204.
| k | k² | Compared to 45 |
|---|---|---|
| 4 | 16 | 29 below |
| 5 | 25 | 20 below |
| 6 | 36 | 9 below |
| 7 | 49 | 4 above |
| 8 | 64 | 19 above |
| 9 | 81 | 36 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 6), divide 45 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 45 ÷ x | New guess (x + 45÷x) ÷ 2 |
|---|---|---|---|
| 1 | 6 | 7.5 | 6.75 |
| 2 | 6.75 | 6.666667 | 6.708333 |
| 3 | 6.708333 | 6.708075 | 6.708204 |
| 4 | 6.708204 | 6.708204 | 6.708204 |
After 4 passes the guess is 6.708204, already matching √45 = 6.708204 to six decimal places.
Exact form from the prime factors
45 = 3^2 × 5. Each pair of matching primes comes out of the root as one factor, and what's left stays inside: √45 = 3√5. That's the exact value; the decimal 6.708204 is a rounding of it.
Frequently Asked Questions
What is the square root of 45?
The square root of 45 is approximately 6.708204, or exactly 3√5 in simplified radical form.
Is 45 a perfect square?
No, 45 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √45 between?
Between 6 and 7, because 6² = 36 and 7² = 49.
What are the square roots of the numbers right next to 45?
√44 is about 6.6332, and √46 is about 6.7823. Square roots increase more slowly as the number gets bigger, which is why consecutive whole numbers have roots that get closer together the higher you go.
What is √45 squared?
Exactly 45 -- squaring undoes the square root. Squaring the rounded decimal 6.708204 gives 45, which is why the radical form 3√5 is the exact answer and the decimal is an approximation.
What is the negative square root of 45?
−6.708204. Every positive number has two square roots, one positive and one negative, because (−6.708204)² is also 45. The radical sign √ by convention means the positive one.
More Number Theory
- Square Root Calculator (look up any number)
- Factors of 45
- GCF Calculator
- LCM Calculator