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 mini-lesson, 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

• 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.

• 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.

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 = 3² × 7²

Step 2. Group the prime factors of 441 in pairs.
441 = 3² × 7²

Step 3. Pick one factor from each pair and the square root of 441 can be written as:
√441 = √(3² × 7²)
√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.

• Square root of 44
• Square root of 14
• Square root of 288
• Square root of 144
• Square root of 400

• 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.

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: 3² × 7²

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.