Square Root Tricks

The value that, when multiplied by itself, results in the original number is known as the square root of that number. The square root symbol is √. The square root of a number can be rational (like 6 or 2.5) or irrational (√2 or √5). If the square root of a number is a whole number, then that number is called a perfect square. For example, √25 is 5.
 

The square root of a perfect square number can be found using basic tricks, without relying on long division. One of the helpful methods is to remember the unit digits of the squares for the first ten natural numbers. These tips can help us find the square root of a number when solving complex problems. Given below is a unit digit table that is used to find the square root of the given number.
 

Finding a square root for a larger number like 1,567,865 can be difficult. There are some simple steps, as given below, that can help you solve them easily. 

Step 1: Starting from the right, group the digits of the number into pairs of two.

Step 2: Look at the last digit of the number. Use the unit digit table of numbers from 1 to 9 to find which numbers can give that last digit when squared. Those are the possible unit digits of the square root.

Step 3: Look at the leftmost pair, i.e., the first pair, and find between which two squares this number lies.

Step 4: The smaller of those two is the tens' digit of the square root.

Step 5: If the unit digit is 5 or 0, then it means the number that we’re looking for also ends with either 5 or 0. If the unit digit is not 0 or 5, use trial and error with the possibilities from Step 3 to find the correct unit digit.

Step 6: Try different digits to complete the divisor and multiply. Choose the largest possible digit that, when used, gives a product less than or equal to the number you're dividing in that step. 
 

The square root of a 3-digit number is usually a 2-digit number, but if the number is a perfect square above 961 (like 1024), the square root can be a 3-digit number. The trick to finding the square root of a three-digit number is given below with a simple example:

Step 1: Group the digits from the right and make pairs of digits. For 729, we group it as 7 | 29 — the rightmost two digits form a pair, and the remaining digit stands alone.

Step 2: Look at the last digit of the number and use the unit table to find the unit digit of the square root of the given number. Here, the last digit is 9, so we have to find the squares that end in 9. 
32 = 9
72 = 49
So, the unit digit of the square root is either 3 or 7.

Step 3: Now, look at the leftmost group of digits and find the largest perfect square less than or equal to that number. The square root of that perfect square gives the tens digit of the final answer. Here, the first digit is 7. The perfect squares which are closest to 7 are:
22 = 4
32 = 9
9 is too big so we are using 22 = 4, and √4 = 2.
So, the tens digit is 2.

Step 4: Combine the two guesses; the number we’re looking for can be one of those two guesses.
The possible guesses are:
23 (tens digit 2, unit digit 3)
27 (tens digit 2, unit digit 7)
Now check both:
23 × 23 = 529
27 × 27 = 729

So, the square root of 729 is ±27.
 

Finding the square root of a 4-digit number follows similar steps as for 3-digit numbers, with a few extra rules due to the larger size. Here's how to do it:

Step 1: Group the digits
Start from the right and group the digits into pairs of two. 

Step 2: Find the nearest perfect square
Look at the first pair and find the biggest perfect square that is less than or equal to it.

Step 3: Find the first digit
Take the square root of that perfect square, and it is the first digit of the answer.

Step 4: Find the remaining digits
Now look at the second pair of digits and find the next digit of the square root.

Step 5: Combine the digits
Put the digits from step 3 and step 4 together to make the square root guess.

Step 6: Check the answer
Multiply the guesses by itself to see whether it gives the original number. 

Example: Find the square root of 9604
Step 1: Group the digits (96)(04)
Step 2: The first pair is 96. The perfect square less than 96 is 81.
Step 3: The square root of 81 is ±9. So, the first digit is 9.
Step 4: Now check the next pair: 04
The unit digits are:
22 = 4
82 = 64
So the second digit is either 2 or 8.
Step 5: Combine the terms: 
92 × 92 = 8464
98 × 98 = 9604
So the square root of 9604 is ±98.
Step 6: Final check: 98 × 98 = 9604
 

Finding the square root of a 5-digit number is difficult, but by using the following trick, you can find it easily.

Step 1: Group the numbers
Start from the right and split the numbers into two parts. The first part should contain 3 digits, and the second part should have 2 digits. 
Example: 53824 can be grouped as | 5 | 38 | 24

Step 2: Check the last digit
Look for the last digit, here it is 4.
Check the numbers that give that number when squared.
22 = 4
82 = 64
So the last digit of the square root is either 2 or 8.

Step 3: Use the first three digits
Look at the first two groups: 538
Find the closest perfect squares
232 = 529
242 = 576
So, 538 is between 529 and 576. We chose 23 as the first part of the answer, as 242 is big.

Step 4: Combine the digits
Now square the number you found to check if it matches the original number.
232 × 232 = 53824
238 × 238 = 56644
So, the square root of 53824 is ±232.
 

Learning the square root tricks helps us calculate faster, even without a calculator. From measuring things to solving puzzles, square root tricks play an important role in both everyday life and advanced fields like design, science, and engineering. Here are some of the real-life applications of square root tricks.

• Sports fields: Players and coaches use square roots to figure out the side length of a square field. If a square playing zone has an area of 625 sq. ft, the length of each side is √625 = 25 ft.
• Maps and navigation: Square root tricks are used in the Pythagorean theorem to find straight line distances between two points, like on maps or GPS. For example, Distance = x2 + y2 (e.g., 32 + 42  = 25 = 5 units).
• Physics and engineering calculations: Square roots appear in formulas for speed, energy, force, and more. Square root tricks can help in solving them faster. For example, to find the speed from kinetic energy:
  v = 2Em (e.g., E = 100 J, m = 2 kg  v = 100 = 10 m/s)
• Video games and computer graphics: Square roots are used in graphics, distance between game characters, or size scaling. Game developers use square root calculations to detect how far a player is from a target.