Square Root of 441 — Value, Simplification, and Steps

Q: Is 441 a perfect square?

Yes. $441 = 21^2$, so its square root is the whole number 21.

Q: What is the prime factorisation of 441?

$441 = 3^2 \times 7^2$, and halving the exponents gives $3 \times 7 = 21$.

Q: What is the square root of 484?

$22$, since $22^2 = 484$, the next perfect square above 441.

Q: What is the square root of −441?

There is no real square root of a negative number; $\sqrt{-441} = 21i$, where $i = \sqrt{-1}$ is the imaginary unit.

The square root of 441 is 21. Because 21 is a whole number, 441 is a perfect square.

Quick Answer:
Result: $\sqrt{441} = 21$
Notation: whole number, $21$
Method shown: prime factorisation and long division
Approximate value: exact — no approximation needed ( $21.000$ )
Exact form: $21$ (441 is a perfect square, $21^2 = 441$ )

Quick Reference Table

Expression	Value	Perfect Square?
$\sqrt{400}$	$20$	Yes ( $20^2$ )
$\sqrt{441}$	$21$	Yes ( $21^2$ )
$\sqrt{484}$	$22$	Yes ( $22^2$ )
$\sqrt{529}$	$23$	Yes ( $23^2$ )
$\sqrt{576}$	$24$	Yes ( $24^2$ )
$441^2$	$194{,}481$	—

Where the Square Root of 441 Appears

$\sqrt{441} = 21$ is the side length of a square whose area is 441 square units, so a 21-by-21 arrangement holds exactly 441 cells. It also turns up in the Pythagorean triple $(20, 21, 29)$ , where $20^2 + 21^2 = 400 + 441 = 841 = 29^2$ , so 21 is a whole-number leg in a right triangle with a whole-number hypotenuse.

What a Perfect Square Is

A perfect square is any integer formed by multiplying an integer by itself: $100 = 10^2$ , $144 = 12^2$ , and here $441 = 21^2$ . Its square root is therefore a whole number with no decimal tail. Because 441 is a perfect square, $\sqrt{441}$ is rational and exact, unlike a root such as $\sqrt{78}$ , which never terminates.

Is the Square Root of 441 Rational or Irrational?

The square root of 441 is rational. It equals the whole number 21, which can be written as the ratio $\frac{21}{1}$ , so it meets the definition of a rational number. Every perfect square has a rational square root.

How to Compute the Square Root of 441

Method 1: Prime Factorisation

Break 441 into primes.

$441 = 3 \times 147$

$= 3 \times 3 \times 49$

$= 3^2 \times 7^2$

Take one factor from each pair (each even power halves):

$\sqrt{441} = \sqrt{3^2 \times 7^2} = 3 \times 7 = 21$

Final answer: $\sqrt{441} = 21$ .

Method 2: Long Division

Pair the digits from the right: $4 \mid 41$ .

Largest square $\le 4$ is $4 = 2^2$ ; first digit 2, remainder 0.

Bring down $41$ to make 41; double the 2 to get 4, and find a digit $d$ so that $4d \times d \le 41$ .

Here $41 \times 1 = 41$ exactly, so the next digit is 1, remainder 0.

Final answer: $\sqrt{441} = 21$ .

Method 3: Estimation

$20^2 = 400$ and $22^2 = 484$ , so $\sqrt{441}$ lies between 20 and 22.

Since 441 ends in 1, its root ends in 1 or 9; the value between 20 and 22 is 21.

Final answer: $\sqrt{441} = 21$ .

Common Mistakes With Square Root of 441

Mistake 1: Taking the whole exponent instead of half

Where it slips in: using prime factorisation, then multiplying the primes as they stand.

Don't do this: write $\sqrt{441} = 3^2 \times 7^2 = 9 \times 49$ .

The correct way: a square root halves each exponent, giving $3^1 \times 7^1 = 21$ .

Mistake 2: Mis-pairing digits in long division

Where it slips in: grouping the digits from the left as $44 \mid 1$ .

Don't do this: pair from the left.

The correct way: always pair from the right: $4 \mid 41$ . Wrong pairing throws off every later digit.

Mistake 3: Forgetting the negative root when solving an equation

Where it slips in: solving $x^2 = 441$ and reporting only 21.

Don't do this: write $x = 21$ as the sole solution.

The correct way: an equation gives $x = \pm 21$ , even though the principal square root symbol $\sqrt{441}$ means the positive value 21 alone.

Where to Go From Here

Confirm that $484 = 22^2$ and $400 = 20^2$ by the prime-factorisation method, 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 classes online.

Was this article helpful?

Your feedback helps us write better content

Frequently Asked Questions

Is 441 a perfect square?

Yes.

441 = 21^2

, so its square root is the whole number 21.

What is the prime factorisation of 441?

441 = 3^2 \times 7^2

, and halving the exponents gives

3 \times 7 = 21

Is the square root of 441 rational or irrational?

Rational — in fact a whole number, 21, because 441 is a perfect square.

What is the square root of 484?

22

, since

22^2 = 484

, the next perfect square above 441.

What is the square root of −441?

There is no real square root of a negative number;

\sqrt{-441} = 21i

, where

i = \sqrt{-1}

is the imaginary unit.

✍️ Written By

Bhanzu Team

Content Creator and Editor

Bhanzu’s editorial team, known as Team Bhanzu, is made up of experienced educators, curriculum experts, content strategists, and fact-checkers dedicated to making math simple and engaging for learners worldwide. Every article and resource is carefully researched, thoughtfully structured, and rigorously reviewed to ensure accuracy, clarity, and real-world relevance. We understand that building strong math foundations can raise questions for students and parents alike. That’s why Team Bhanzu focuses on delivering practical insights, concept-driven explanations, and trustworthy guidance-empowering learners to develop confidence, speed, and a lifelong love for mathematics.