In math, the square root of a number like 421 is a number that, when multiplied by itself, is equal to 421. We would show this in mathematical form with the square root symbol, which is called the radical symbol: √
Any number with the radical symbol next to it us called the radical term or the square root of 421 in radical form.
To explain the square root a little more, the square root of the number 421 is the quantity (which we call q) that when multiplied by itself is equal to 421:
So what is the square root of 421 and how do we calculate it? Well if you have a computer, or a calculator, you can easily calculate the square root. If you need to do it by hand, then it will require good old fashioned long division with a pencil and piece of paper.
For the purposes of this article, we'll calculate it for you (but later in the article we'll show you how to calculate it yourself with long division). The square root of 421 is 20.518284528683:
Is 421 a Perfect Square?
When the square root of a given number is a whole number, this is called a perfect square. Perfect squares are important for many mathematical functions and are used in everything from carpentry through to more advanced topics like physics and astronomy.
If we look at the number 421, we know that the square root is 20.518284528683, and since this is not a whole number, we also know that 421 is not a perfect square.
If you want to learn more about perfect square numbers we have a list of perfect squares which covers the first 1,000 perfect square numbers.
Is 421 a Rational or Irrational Number?
Another common question you might find when working with the roots of a number like 421 is whether the given number is rational or irrational. Rational numbers can be written as a fraction and irrational numbers can't.
The quickest way to check if a number is rational or irrational is to determine if it is a perfect square. If it is, then it's a rational number, but if it is not a perfect square then it is an irrational number.
We already know that 421 is not a rational number then, because we know it is not a perfect square.
Calculating the Square Root of 421
To calculate the square root of 421 using a calculator you would type the number 421 into the calculator and then press the √x key:
To calculate the square root of 421 in Excel, Numbers of Google Sheets, you can use the SQRT()
function:
Rounding the Square Root of 421
Sometimes when you work with the square root of 421 you might need to round the answer down to a specific number of decimal places:
10th: √421 = 20.5
100th: √421 = 20.52
1000th: √421 = 20.518
Finding the Square Root of 421 with Long Division
If you don't have a calculator or computer software available, you'll have to use good old fashioned long division to work out the square root of 421. This was how mathematicians would calculate it long before calculators and computers were invented.
Step 1
Set up 421 in pairs of two digits from right to left and attach one set of 00 because we want one decimal:
4

21

00

Step 2
Starting with the first set: the largest perfect square less than or equal to 4 is 4, and the square root of 4 is 2 . Therefore, put 2 on top and 4 at the bottom like this:
2  
4

21

00

4



Step 3
Calculate 4 minus 4 and put the difference below. Then move down the next set of numbers.
2  
4

21

00

4



0

21


Step 4
Double the number in green on top: 2 × 2 = 4. Then, use 4 and the bottom number to make this problem:
4? × ? ≤ 21
The question marks are "blank" and the same "blank". With trial and error, we found the largest number "blank" can be is 0. Replace the question marks in the problem with 0 to get:
40 × 0 = 0
Now, enter 0 on top, and 0 at the bottom:
2  0  
4

21

00

4



0

21


0



Step 5
Calculate 21 minus 0 and put the difference below. Then move down the next set of numbers.
2  0  
4

21

00

4



0

21


0



0

21

00

Step 6
Double the number in green on top: 20 × 2 = 40. Then, use 40 and the bottom number to make this problem:
40? × ? ≤ 2100
The question marks are "blank" and the same "blank". With trial and error, we found the largest number "blank" can be is 5.
Now, enter 5 on top:
2  0  5 
4

21

00

4



0

21


0



0

21

00

Hopefully, this gives you an idea of how to work out the square root using long division so you can calculate future problems by yourself.
Practice Square Roots Using Examples
If you want to continue learning about square roots, take a look at the random calculations in the sidebar to the right of this blog post.
We have listed a selection of completely random numbers that you can click through and follow the information on calculating the square root of that number to help you understand number roots.
Calculate Another Square Root Problem
Enter your number in box A below and click "Calculate" to work out the square root of the given number.