Square Root of 3600 — Value & Steps
Square Root of 3600 — Value & Steps
TL;DR
The square root of 3600 is exactly 60, because 60×60=3600. This article shows why 3600 is a perfect square, works the prime-factorisation and long-division methods, gives a quick-reference table, and points to where 3600 appears.
The square root of 3600 is 60. Because 60 is a whole number, 3600 is a perfect square.
Quick Answer:
Result: 3600=60
Notation: whole number, 60
Method shown: prime factorisation and long division
Approximate value: exact — no approximation needed (60.000)
Exact form: 60 (3600 is a perfect square, 60² = 3600)
Quick Reference Table
| Expression | Value | Perfect Square? |
|---|---|---|
| 2500 | 50 | Yes (50² = 2500) |
| 3025 | 55 | Yes (55² = 3025) |
| 3600 | 60 | Yes (60² = 3600) |
| 4225 | 65 | Yes (65² = 4225) |
| 4900 | 70 | Yes (70² = 4900) |
| 3600² | 12,960,000 | — |
Where the Square Root of 3600 Appears
3600=60 is the side length of a square whose area is 3600 square units, so a 60-by-60 grid holds exactly 3600 cells. The number 3600 is also the count of seconds in one hour, that is 60×60, which is exactly why its square root is a whole number: 3600 was built as 60² by the base-60 timekeeping the Babylonians handed down.
What a Perfect Square Is
A perfect square is any integer formed by multiplying an integer by itself: 900=30², 2500=50², and here 3600=60². Its square root is therefore a whole number with no decimal tail. Because 3600 is a perfect square, ( \sqrt{3600} ) is rational and exact, unlike a root such as ( \sqrt{78} ), which never terminates.
Is the Square Root of 3600 Rational or Irrational?
The square root of 3600 is rational. It equals the whole number 60, which can be written as the ratio ( \frac{60}{1} ), so it meets the definition of a rational number. Every perfect square has a rational square root.
How to Compute the Square Root of 3600
Method 1: Prime Factorisation
Break 3600 into primes.
( 3600=36 \times 100 = (2^2 \times 3^2) \times (2^2 \times 5^2) = 2^4 \times 3^2 \times 5^2)
Take one factor from each pair (each even power halves):
3600=( 2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60 )
Final answer: 3600=60.
Method 2: Long Division
Pair the digits from the right: 36∣00.
Largest square ≤36 is 36=6; first digit 6, remainder 0.
Bring down 00 to make 0; double the 6 to get 12, and the next digit is 0 because 120×0=0.
Remainder 0, so the root is 60.
Final answer: 3600=60.
Method 3: Splitting Off the Trailing Zeros
Write 3600=36×100.
3600=36×100=6×10=60.
Final answer: 3600=60.
Common Mistakes With Square Root of 3600
Mistake 1: Halving each trailing zero
Where it slips in: thinking "3600 has two zeros, so drop one and take √36 = 6, giving 6 tens."
Don't do this: the shortcut is fine, but only when the digit block is itself a perfect square.
The correct way: split as 36×100=6×10=60.
Mistake 2: Taking the whole exponent instead of half
Where it slips in: after prime factorisation, multiplying the primes as they stand.
Don't do this: write 3600=2⁴×3²×5².
The correct way: halve each exponent, giving 60.
Mistake 3: Confusing 3600 with 360
Where it slips in: dropping a zero while copying the problem.
Don't do this: report 360≈18.97 as if it were 3600.
The correct way: 3600=60 exactly; 360 is irrational and roughly 18.974, a completely different number.
Frequently Asked Questions
Is 3600 a perfect square?
Yes. 3600=60², so its square root is the whole number 60.
What is the prime factorisation of 3600?
3600=2⁴×3²×5², and halving the exponents gives 60.
Is the square root of 3600 rational or irrational?
Rational — in fact a whole number, 60, because 3600 is a perfect square.
What is the square root of 4900?
70, since 70² = 4900, the next multiple-of-ten square above 3600.
What is the square root of −3600?
There is no real square root of a negative number; −3600=60i, where i=√−1 is the imaginary unit.