Square Root of 4489: Value, Methods (Prime Factorization & Long Division)

The square root of 4489 is 67.

√4489 = 67, because 67 × 67 = 4489.

Property	Value
Number	4489
Square root	67 (and −67)
Perfect square?	Yes
Prime factorization	67²
Rational or irrational?	Rational (a whole number)

If that’s all you needed, you can close the tab happy. But if you’ve got homework to hand in or an exam coming up, you’ll probably need to show your working, and that’s what the rest of this page is for. We’ll go through three ways to get there: a ten-second estimation trick, prime factorization and long division. All three end at 67, so you can use one to check the other.

☰ Table of Contents

What Is a Square Root? Understanding the Square Root of 4489

A square root is the number that gives you the original number when you multiply it by itself. That’s really all it is.

√49 = 7, because 7 × 7 = 49
√144 = 12, because 12 × 12 = 144
√4489 = 67, because 67 × 67 = 4489

There’s one little twist that catches a lot of students out. Every positive number has two square roots, one positive and one negative. (−67) × (−67) also gives 4489, so both 67 and −67 work. When you see the √ symbol on its own, though, it means the positive one, which we call the principal square root. So √4489 = 67.

Is 4489 a Perfect Square?

It is. A perfect square is a number you get by multiplying a whole number by itself, and 4489 = 67 × 67.

Here’s the fun part: you can often guess this before doing any real work.

It ends in 9. A perfect square ending in 9 always has a root ending in 3 or 7. And 67 ends in 7, so that’s a good sign.
It sits between two squares you probably know. 66² = 4356 and 68² = 4624, so 4489 lands right in between.
Its digit sum is 25 (4 + 4 + 8 + 9), which reduces to 7 (2 + 5). Perfect squares always have a digital root of 1, 4, 7 or 9, so 7 passes the test.

None of these proves it by itself. Together, though, they tell you it’s worth checking, and the methods below do the proving.

Is 4489 a Perfect Square? Quick Checks Before Finding it

This is the one I always show students first, because it feels a bit like magic. It’s perfect for multiple-choice exams, where you don’t need the working, just the right option.

Step 1: Box the number in. 60² = 3600 and 70² = 4900. Since 4489 sits between them, the root has to be somewhere between 60 and 70.

Step 2: Look at the last digit. 4489 ends in 9, so the root must end in 3 or 7. That leaves just two candidates: 63 and 67.

Step 3: Pick the winner. Try 65 as a halfway point. 65² = 4225, which is smaller than 4489. So the root must be bigger than 65, and 63 is out.

Answer: 67. Quick check: 67 × 67 = 4489. ✓

Three small steps, no long calculation, and you’ve got your answer.

Square Root of 4489 by Prime Factorization

This method breaks 4489 down into prime numbers, then pairs them up. It’s brilliant for some numbers, and a bit of a slog for this one. I’ll show you why.

Step 1: Decide where to stop. You only need to test primes up to the square root of the number. Since √4489 is about 67, that’s where our list ends.

Step 2: Test the primes one by one.

2, 3 and 5: None of them work. 4489 is odd, its digit sum (25) isn’t a multiple of 3, and it doesn’t end in 0 or 5.
7: 7 × 641 = 4487, so the remainder is 2.
11: The alternating digit sum is 4 − 4 + 8 − 9 = −1, which isn’t a multiple of 11.
13: 13 × 345 = 4485, remainder 4.
17: 17 × 264 = 4488, remainder 1. So close, but no.
19: 19 × 236 = 4484, remainder 5.
23: 23 × 195 = 4485, remainder 4.
29, 31, 37, 41 and 43: Remainders of 23, 25, 12, 20 and 17. None divide evenly.
47, 53, 59 and 61: Remainders of 24, 37, 5 and 36. Still nothing.
67: 67 × 67 = 4489. There it is. ✓

Step 3: Write out the factorization. 4489 = 67 × 67 = 67²

Step 4: Pair the factors. There’s exactly one pair of 67s, so one 67 steps out of the root:

√4489 = √(67 × 67) = 67

Honestly, that was the long way round. You had to rule out nearly twenty primes before finding 67. Prime factorization really shines when the factors are small, like 144 = 2⁴ × 3². For a number like 4489, the next method is much quicker.

Square Root of 4489 by Long Division Method

If a number doesn’t have small factors, this is the method most teachers and exam papers expect. It builds the answer one digit at a time, and it keeps working even on numbers that aren’t perfect squares.

Step 1: Pair the digits from the right. 4489 becomes 44 | 89.

Step 2: Find the first digit. What’s the biggest number whose square fits into 44? 6² = 36 fits, but 7² = 49 is too big. So the first digit is 6. Write 6 as the first digit of the answer and as the first divisor, then subtract: 44 − 36 = 8.

Step 3: Bring down the next pair. Bring down 89 next to the 8 to make 889.

Step 4: Double the quotient. Double 6 to get 12. Your new divisor is 12_, with a blank at the end waiting for a digit.

Step 5: Find the missing digit. You need a digit d so that (120 + d) × d fits into 889. Let’s try a few:

d = 6: 126 × 6 = 756 (fits, but there’s room for a bigger digit)
d = 7: 127 × 7 = 889 (a perfect fit!)
d = 8: 128 × 8 = 1024 (too big)

So d = 7.

Step 6: Subtract. 889 − 889 = 0. A zero remainder tells you the number is a perfect square, and you’re done.

Here’s what it looks like on paper:

6 7

______

6 | 44 89

—-

127| 8 89

8 89

—-

√4489 = 67 ✓

Which Method Should You Use?

Method	Best for	Speed for 4489
Estimation	Multiple-choice exams	Fastest
Prime factorization	Numbers with small prime factors	Slow (67 is a large prime)
Long division	Large numbers, decimals, non-perfect squares	Quick and reliable

My honest advice: use estimation to get the answer fast, and long division to show your working. Keep prime factorization for numbers like 576 or 1764, where the factors are small and friendly.

A Few Fun Facts About 4489

It’s the square of a prime. 67 is prime, so 4489 has exactly three factors: 1, 67 and 4489.
It’s not a perfect cube. Its cube root is about 16.5, which isn’t a whole number.
Its neighbors follow a neat pattern. 66² = 4356, and the gap from there to 4489 is 133, which equals 2 × 66 + 1. That’s the rule for consecutive squares: n² + 2n + 1 = (n + 1)².

Mistakes Almost Everyone Makes

I’ve seen these over and over, so don’t feel bad if you’ve made one too.

1. Pairing the digits wrongly. 4489 pairs as 44 | 89, not 4 | 48 | 9. For whole numbers, always pair from the right.
2. Choosing 7 as the first digit. 7² = 49 is bigger than 44, so the first digit has to be 6.
3. Forgetting to double the quotient. The divisor starts from 2 × 6 = 12, not 6.
4. Using different digits in different places. The digit in the divisor, the digit you multiply by and the digit in the answer must all match. Here, that’s 7.
5. Testing primes beyond the square root. You never need to go past 67 here.
6. Forgetting the negative root. If a question asks you to solve x² = 4489, the answer is x = 67 and x = −67.

Worksheet for Square Root

Want to make sure it’s really stuck? Give these a go:

√4624 (answer: 68)
√4356 (answer: 66)
√5329 (answer: 73)

For each one, box it in between two squares you know, use the last digit to narrow it down, then confirm with a long division. Do all three and you’ll have the routine down.

Conclusion

√4489 = 67
4489 is a perfect square, equal to 67 × 67
Estimation is the fastest route: box it in (60² to 70²), then use the last digit (3 or 7) to land on 67
Long division confirms it in two moves: 6 first, then 7
Prime factorization works too, but it’s slow here because 67 is a fairly large prime
Always check by multiplying: 67 × 67 = 4489

Once you’ve done this one by hand, the next square root you meet won’t feel nearly as scary.

Frequently Asked Questions

What is the square root of 4489?

It’s 67, because 67 × 67 = 4489.

Is 4489 a perfect square?

Yes. It equals 67², and 67 is a whole number.

What is the prime factorization of 4489?

4489 = 67 × 67 = 67². Since 67 is prime, it can’t be broken down any further.

Is the square root of 4489 rational or irrational?

Rational. It’s the whole number 67, which can be written as 67/1.

What is the negative square root of 4489?

−67, because (−67) × (−67) = 4489. The √ symbol on its own means the positive root.