The square root of 3600 is 60. Because 60 is a whole number, 3600 is a perfect square.
Quick Answer:
Result: $\sqrt{3600} = 60$
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, $60^2 = 3600$)
Quick Reference Table
Expression | Value | Perfect Square? |
|---|---|---|
$\sqrt{2500}$ | $50$ | Yes ($50^2$) |
$\sqrt{3025}$ | $55$ | Yes ($55^2$) |
$\sqrt{3600}$ | $60$ | Yes ($60^2$) |
$\sqrt{4225}$ | $65$ | Yes ($65^2$) |
$\sqrt{4900}$ | $70$ | Yes ($70^2$) |
$3600^2$ | $12{,}960{,}000$ | — |
Where the Square Root of 3600 Appears
$\sqrt{3600} = 60$ 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 $60 \times 60$, which is exactly why its square root is a whole number: 3600 was built as $60^2$ 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: $900 = 30^2$, $2500 = 50^2$, and here $3600 = 60^2$. 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):
$\sqrt{3600} = 2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60$
Final answer: $\sqrt{3600} = 60$.
Method 2: Long Division
Pair the digits from the right: $36 \mid 00$.
Largest square $\le 36$ is $36 = 6^2$; first digit 6, remainder 0.
Bring down $00$ to make 0; double the 6 to get 12, and the next digit is 0 because $120 \times 0 = 0$.
Remainder 0, so the root is $60$.
Final answer: $\sqrt{3600} = 60$.
Method 3: Splitting Off the Trailing Zeros
Write $3600 = 36 \times 100$.
$\sqrt{3600} = \sqrt{36} \times \sqrt{100} = 6 \times 10 = 60$
Final answer: $\sqrt{3600} = 60$.
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 $\sqrt{36} \times \sqrt{100} = 6 \times 10 = 60$, but only because each factor is itself a perfect square before you split.
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 $\sqrt{3600} = 2^4 \times 3^2 \times 5^2$.
The correct way: halve each exponent, giving $2^2 \times 3 \times 5 = 60$.
Mistake 3: Confusing 3600 with 360
Where it slips in: dropping a zero while copying the problem.
Don't do this: report $\sqrt{360} \approx 18.97$ as if it were $\sqrt{3600}$.
The correct way: $\sqrt{3600} = 60$ exactly; $\sqrt{360}$ is irrational and roughly 18.974, a completely different number.
Where to Go From Here
Confirm that $2500 = 50^2$ and $4900 = 70^2$ by splitting off the trailing zeros, then compare them with the perfect squares in the table above. To build these skills with a teacher, explore Bhanzu's algebra tutor or math tutoring.
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