Square Root of 441
The number 441 is an odd composite number. The square root of 441 is the number (integer) which when multiplied by itself results in 441, it is both positive and negative. In this minilesson, we will find the square root of 441 by prime factorization and long division method, learn a few interesting facts, and solve some problems.
 Square root of 441: 21
 Square of 441: 194,481
1.  What Is the Square Root of 441? 
2.  Is Square Root of 441 Rational or Irrational? 
3.  How to Find the Square Root of 441? 
4.  FAQs on Square Root of 441 
What Is the Square root of 441?
 The square root of a number is a number (Integer) whose product by itself results in the initial number.
 441 = a × a = 21 × 21
 Then a = √441 = √(21 × 21)
 21 × 21 = 441 or 21 × 21 = 441
 The square root of 441 is +21 or 21.
 This shows that 441 is a perfect square.
Is Square Root of 441 Rational or Irrational?
 A rational number is a number that can be written in the ratio of two integers, that is, p/q where q ≠ 0.
 21 and 21 can be written as 21/1 and 21/1
 So, the square root of 441 is a rational number.
How to Find the Square Root of 441?
The square of 441 can be evaluated using the prime factorization method, long division method, or the repeated subtraction method.
Square Root of 441 by Prime Factorization Method
The following steps can be followed to find the square root of 441 using prime factorization:
Step 1. Determine the prime factorization of 441.
441 = 3 × 3 × 7 × 7
441 = 3^{2} × 7^{2}
Step 2. Group the prime factors of 441 in pairs.
441 = 3^{2} × 7^{2}
Step 3. Pick one factor from each pair and the square root of 441 can be written as:
√441 = √(3^{2} × 7^{2})
√441 = √(3 × 7)^{2}
√441 = ((3 × 7)^{2})^{1/2} = ±(3 × 7)
√441 = ±21
Square Root of 441 by Long Division
Follow the steps shown below to find the square root of 441 by long division.
Step 1. Write 441 as shown in the figure. Pair the number from the right end (1) by putting a bar on top of them. In the case of 441, 41 will be paired under one bar and 4 under the second bar.
Step 2. Find a number that when multiplied by itself results in a number equal to or less than 4.
Step 3. Drag the next pair of numbers. Here it is 41.
Multiply the quotient 2 by 2 (or add with itself) and write it as the new divisor’s ten’s place digit.
Step 4. Choose a number for the unit’s place of the divisor in such a way that when it is multiplied with the new divisor, it results in 41 or a smaller number closest to 41.
Here the number is 1 as 41 × 1 = 41.
So, we get the value of the square root of √441 = 21 by the long division method.
Explore square roots using illustrations and interactive examples.
Important notes:
 There will be n/2 digits in the square root of an even number with n digits.
 There will be (n+2)/2 digits in the square root of an odd number with n digits.
 The number 441 is a perfect square.
Square Root of 441 Solved Examples

Example 1: Peter wants to find the square root of the reverse of 441, which is 144. Can you help Peter?
Solution:
Prime factorization of 144: 2^{4} × 3^{2}
Therefore, the square root of 144 is 2 × 2 × 3 = 12
Hence, the square root of 144 is 12 which also the reverse of the square root of 441 (21). 
Example 2: Find the square root of 441 by repeated subtraction method?
Solution:
Sr. No.
Subtraction
Sr. No.
Subtraction
1.
441  1 = 440
12.
320  23 = 297
2.
440  3 = 437
13.
297  25 = 272
3.
437  5 = 432
14.
272  27 = 245
4.
432  7 = 425
15.
245  29 = 216
5.
425  9 = 416
16.
216  31 = 185
6.
416  11 = 405
17.
185  33 = 152
7.
405  13 = 392
18.
152  35 = 117
8.
392  15 = 377
19.
117  37 = 80
9.
377  17 = 360
20.
80  39 = 41
10.
360  19 = 341
21.
41  41 = 0
11.
341  21 = 320
So, starting from 441, we have subtracted 21 times to get 0.
Thus the square root of 441 is 21.
FAQs on Square Roots of 441
Is the square root of 441 a rational number?
Yes, the square root of 441 is a rational number.
The square roots of 441 are 21 and 21, which are rational numbers.
What is the prime factorization of 441?
Prime factorization of 441: 3^{2} × 7^{2}
By what different methods can we find the square root of 441?
We can find the square root of 441 by the following methods:
 Long division method
 Prime factorization
 Repeated subtraction method
What is the square root of 441?
The square root of negative numbers is imaginary.
It is represented as √441 = 21i.
How is the square root of 441 is represented in exponential form?
The square root of 441 is represented as (441)^{1/2} in exponential form.
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