# Square Root of 441 — Value, Simplification, and Steps

TL;DR

The square root of 441 is exactly 21, because 21×21=441. This article shows why 441 is a perfect square, works the prime-factorisation and long-division methods, gives a quick-reference table, and points to where \( \sqrt{441} \) appears.

Last updated on July 18, 2026

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 \( 20^2 + 21^2 = 841 = 29^2 \), 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∣41.  
Largest square ≤4 is 4 = 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×d≤41.
Here 41×1=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.  
**Don't do this:** pair from the left.  
**The correct way:** always pair from the _right_.

### **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 \( x = \pm 21 \).

## Where to Go From Here

Confirm that \( 484 = 22^2 \) and \( 400 = 20^2 \) by the prime-factorisation method. To build these skills with a teacher, explore Bhanzu's [algebra tutor](/content/math/tutor/algebra/index.html) or [math classes online](/content/math/classes/index.html).

## 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 \( i = \sqrt{-1} \).

**Written By**: Bhanzu Team  
Content Creator and Editor.