Competitive exams have at least one question where you have to find the square root of a number in seconds. With the help of this shortcut method, you can find square root of a number within seconds. This method is applicable only for perfect squares. The Square root questions asked in competitive exams/bank exams & entrance exams are usually perfect squares. Refer How to check if a number is a perfect square? to find out if a given number is a perfect square.
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Shortcut to find Square Root
Shortcut to find Square Root of any Number
In every competitive exam, there is at least one instance where you will have to find the square root of a number quickly. With the help of this shortcut on how to find the square root of a number, you will be able to find out the square root of any number within seconds. This method is applicable only for perfect squares. Square root questions asked in competitive exams/bank exams are usually perfect squares. Refer How to check if a number is a perfect square? to find out if a given number is a perfect square.
Steps to Find the Square Root of a Number
Step 1: First of all group the number in pairs of 2 starting from the right.
Step 2: To get the ten’s place digit, Find the nearest square (equivalent or greater than or less than) to the first grouped pair from left and put the square root of the square.
Step 3: To get the unit’s place digit of the square root
Remember the following
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If the number ends in
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Unit’s place digit of the square root
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1
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1 or 9(10-1)
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4
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2 or 8(10-2)
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9
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3 or 7(10-3)
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6
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4or 6(10-4)
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5
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5
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0
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0
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Let’s see the logic behind this method to find square root for a better understanding
We know,
12=1
22=4
32=9
42=16
52=25
62=36
72=49
82=64
92=81
102=100
Now, observe the unit’s place digit of all the squares.
Do you find anything common?
We notice that,
Unit’s place digit of both 12and 92 is 1.
Unit’s place digit of both 22 and 82 is 4
And Unit’s place digit of both 32 and 72 is 9
Unit’s place digit of both 42 and 62 is 6.
Step 4: Multiply the ten’s place digit (found in step 1) with its consecutive number and compare the result obtained with the first pair of the original number from left.
Remember,
If the first pair of the original number > Result obtained on multiplication then select the greater number out of the two numbers as the unit’s place digit of the square root.
If the first pair of the original number < the result obtained on multiplication, then select the lesser number out of the two numbers as the unit’s place digit of the square root.
Let us consider an example to get a better understanding of the method
Shortcut to find the square root of a 3 Digit Number
Example 1: √784=?
Step 1: We start by grouping the numbers in pairs of two from the right as follows
7 84
Step 2: To get the ten’s place digit,
We find that nearest square to the first group (7) is 4 and √4=2
Therefore, ten’s place digit=2.
Step 3: To get the unit’s place digit,
We notice that the number ends with 4, So the unit’s place digit of the square root should be either 2 or 8(Refer table).
Step 4: Multiplying the ten’s place digit of the square root that we arrived at in step 1(2) and its consecutive number(3) we get,
2 x 3=6
ten’s place digit of original number > Multiplication result
7>6
So we need to select the greater number (8) as the unit’s place digit of the square root.
Unit’s place digit =8
Ans:√784=28
Shortcut to find the square root of any 4 digit number
Example 2: √3721?
Step 1: We start by grouping the numbers in pairs of two from the right as follows
37 21
Step 2: To get the ten’s place digit,
We find that nearest square to the first group (37)is 36 and√36=6
Therefore ten’s place digit=6
Step 3: To get the unit’s place digit,
We notice that the number ends with 1, So the unit’s place digit of the square root should be either 1 or 9(Refer table).
Step 4: Multiplying the ten’s place digit of the square root that we arrived at in step 1(6) and its consecutive number(7) we get,
6 x 7=42
ten’s place digit of an original number<Multiplication result
37 < 42
So we need to select the lesser number (1) as the unit’s place digit of the square root.
Unit’s place digit =1
Ans:√3721=61
How to find square root shortcut
Example 3: √6889?
Step 1: We start by grouping the numbers in pairs of two from the right as follows
68 89
Step 2: To get the ten’s place digit,
We find that nearest square to the first group (68)is 64 and√64=8
Therefore ten’s place digit=8
Step 3: To get the unit’s place digit,
We notice that the number ends with 9, So the unit’s place digit of the square root should be either 3 or 7(Refer table).
Step 4: Multiplying the ten’s place digit of the square root that we arrived at in step 1(8) and its consecutive number(9) we get,
8 x 9 =72
ten’s place digit of an original number<Multiplication result
68 < 72
So we need to select the lesser number (3) as the unit’s place digit of the square root.
Unit’s place digit =3
Ans:√6889=83
Shortcut to find the square root of any 5 digit number
Example 4: √64516
Step 1: We start by grouping the numbers in pairs of two from the right as follows
645 16
Step 2: To get the ten’s place digit,
We find that nearest square to the first group (645)is 625 and√625=25
Therefore ten’s place digit=25
Step 3: To get the unit’s place digit,
We notice that the number ends with 6, So the unit’s place digit of the square root should be either 4 or 6(Refer table).
Step 4: Multiplying the ten’s place digit of the square root that we arrived at in step 1(25)and its consecutive number(26)we get,
25 x 26=650
ten’s place digit of an original number<Multiplication result
645< 650
So we need to select the lesser number (4) as the unit’s place digit of the square root.
Unit’s place digit =4
Ans:√64516=254
Shortcut to find the square root of any 6 digit number
Example 5: √126736
Step 1: We start by grouping the numbers in pairs of two from the right as follows. For a 6 digit number, this is how it should be paired.
1267 36
Step 2: To get the ten’s place digit,
We find that nearest square to the first group (1267)is 1225 and√1225=35
Therefore ten’s place digit=35
Step 3: To get the unit’s place digit,
We notice that the number ends with 6, So the unit’s place digit of the square root should be either 4 or 6(Refer table).
Step 4: Multiplying the ten’s place digit of the square root that we arrived at in step 1(35)and its consecutive number(36)we get,
35 x 36=1260
ten’s place digit of original number>Multiplication result
1267>1260
So we need to select the greater number (6) as the unit’s place digit of the square root.
Unit’s place digit =6
Ans:√126736=356
Example 6: √207025
Step 1: We start by grouping the numbers in pairs of two from right as follows. For a 6 digit number, this is how it should be paired.
2070 25
Step 2: To get the tenth place digit,
We find that nearest square to the first group (2070) is 2025 and√2025=45. Therefore ten’s place digit=45
Step 3: To get the unit’s place digit,
We notice that the number ends with 5, So the unit’s place digit of the square root should be 5(Refer table).
Skip all the other steps for a number ending in 5
Unit’s place digit =5
Ans:√207025=455
Also, read Square root shortcut Non-perfect squares
Shortcut to Find Cube Root
Shortcut to find cube root is especially helpful in competitive exams where every second counts.
By using this method you can save a lot of time and also get accurate results.
By using this method you can save a lot of time and also get accurate results.
Shortcut to find cube root of a number Mentally
Steps
Step 1:Find the cube root of the last digit.
Points to be remembered while using this method.
(1)If the last digit is 8 then cube root will be 2.
(2)If the last digit is 2 then cube root will be 8.
(3)If the last digit is 7 then cube root will be 3.
(4)If the last digit is 3 then cube root will be 7.
(5)If the last digit is any other digit other than 2,8,3,7 then put the same number.
From this step you will get the unit’s or one’s place digit.
To find the tenth place digit you need to follow the below steps.
Step 2:Strike out the last 3 digits of the given number.
Step3:Find the nearest cube of the remaining number.
Step 4:Find the cube root of the nearest cube which will give you ten’s place digit.
List of cubes to be memorized
13=1 43=64 73=343
23=8 53=125 83=512
33=27 63=216 93=729
Step 4:Find the cube root of the nearest cube which will give you ten’s place digit.
List of cubes to be memorized
13=1 43=64 73=343
23=8 53=125 83=512
33=27 63=216 93=729
NOTE: Shortcut to find cube root of any 5 or 6 digit number is applicable only if the given number is a perfect cube.
You can verify whether the given number is a perfect cube or not by using the Prime Factorization Method. If you are not familiar with this method check it here.
Cube root Shortcut of a 6 digit number
Let me explain this with an example to make things more clear and easy to understand.
So lets say we want to find the cube root of a 6 digit number 15746
Example 1:Find the cube root of 157464 in 5 seconds.
∛157464=?
Step 1:First we need to find the cube root of the last digit of the given number.
Here the last digit is 4 . 4 is a number other than 2, 8, 3, 7. Hence we put the number as it is.We get our one’s place digit as 4.
Now to get tenth place digit
Step 2: We need to strike the last 3 digits of the given number.
In this example the last 3 digits are 464 which we will strike off as shown below
157464
Step 3: We need to find the nearest cube to the remaining number(157).We find that 125 is the nearest cube to 157.
Step 4: We need to find the cube root of the nearest cube(125)
∛125=5
From this step we get our ten’s place digit as 5.
From step 1 and step 4 we get the
∛157464=54
Ans: Cube root of 157464 is 54
Example 2:Find the cube root of 110592 in 5 seconds.
∛110592=?
Step 1:First we need to find the cube root of the last digit of the given number.Here the last digit is 2 . As per table, we will consider 8 as the cube root of 2. We get our one’s place digit as 8.
Now to get tenth place digit
Step 2: We need to strike the last 3 digits of the given number.
In this example the last 3 digits are 592 which we will strike off as shown below
110592
Step 3: We need to find the nearest cube to the remaining number(110).We find that 64 is the nearest cube to 110.
Step 4: We need to find the cube root of the nearest cube(64)
∛64=4
From this step we get our ten’s place digit as 4.
From step 1 and step 4 we get the
∛110592=48
Ans: Cube root of 110592 is 48
Example 3:Find the cube root of 148877 in 5 seconds.
∛148877=?
Step 1:First we need to find the cube root of the last digit of the given number.Here the last digit is 7. As per table, we will consider 3 as the cube root of 7. We get our one’s place digit as 3.
Now to get tenth place digit
Step 2: We need to strike the last 3 digits of the given number.
In this example the last 3 digits are 877 which we will strike off as shown below
148877
Step 3: We need to find the nearest cube to the remaining number(148).We find that 125 is the nearest cube to 148.
Step 4: We need to find the cube root of the nearest cube(64)
∛125=5
From this step we get our ten’s place digit as 5.
From step 1 and step 4 we get the
∛148877=53
Ans:Cube root of 148877 is 53
More Examples
Example 4:Find the cube root of 328509 in 5 seconds.
∛328509=?
Step 1:First we need to find the cube root of the last digit of the given number. Here the last digit is 9.9 is a number other than 2, 8, 3, 7. Hence we put the number as it is.We get our one’s place digit as 9.
Now to get tenth place digit
Step 2: We need to strike the last 3 digits of the given number.
In this example the last 3 digits are 509 which we will strike off as shown below
328509
Step 3: We need to find the nearest cube to the remaining number(328).We find that 216 is the nearest cube to 328.
Step 4: We need to find the cube root of the nearest cube(216)
∛216=6
From this step we get our ten’s place digit as 6.
From step 1 and step 4 we get the
∛328509=69
Ans: Cube root of 328509 is 69
Example 5:Find the cube root of 636056 in 5 seconds.
∛636056=?
Step 1:First we need to find the cube root of the last digit of the given number. Here the last digit is 6.6 is a number other than 2, 8, 3, 7. Hence we put the number as it is.We get our one’s place digit as 6.
Now to get tenth place digit
Step 2: We need to strike the last 3 digits of the given number.
In this example the last 3 digits are 056 which we will strike off as shown below
636056
Step 3: We need to find the nearest cube to the remaining number(636).We find that 512 is the nearest cube to 636.
Step 4: We need to find the cube root of the nearest cube(512)
∛512=8
From this step we get our ten’s place digit as 8.
From step 1 and step 4 we get the
∛636056=86
Ans: Cube root of 636056 is 86
Cube root Shortcut of a 5 digit number
Now, that you have understood how to find cube root of a number within seconds
lets see how this method works for a 5 digit number with the help of an example. Short cut to find Cube Root of 5 digit number is similar to finding cube root of 6 digit number.
Example 1:Find the cube root of 32768
∛32768=?
Step 1: First,we need to find the cube root of the last digit of the given number. Here the last digit is 8.
∛8=2
We get the one’s place digit as 2.
Step 2: We need to strike out the last 3 digits of the given number.
In this example,768 are the last 3 digits which we we will strike off as shown below
32768
Step 3: We find that the nearest cube to the remaining number(32) is 27.
Step 4: Now, we find that the cube root of 27
That will be ∛27=3
From this step we get the ten’s place digit as 3.
Therefore, from step 1 and step 2 we get
∛32768=32
Ans:32
Example 2:Find the cube root of 19683 in 5 seconds.
∛19683=?
Lets go through the steps as below
Step 1:First we need to find the cube root of the last digit of the given number.Here the last digit is 3. As per table, we will consider 7 as the cube root of 3. We get our one’s place digit as 7.
Now to get tenth place digit
Step 2: We need to strike the last 3 digits of the given number.
In this example the last 3 digits are 683 which we will strike off as shown below
19683
Step 3: We need to find the nearest cube to the remaining number(19).We find that 8 is the nearest cube to 19.
Step 4: We need to find the cube root of the nearest cube(64)
∛8=2
From this step we get our ten’s place digit as 2.
From step 1 and step 4 we get the
∛19683=27
Ans:27
More Examples
Example 3:Find the cube root of 29791 in 5 seconds.
∛29791=?
Step 1:First we need to find the cube root of the last digit of the given number. Here the last digit is 1.1 is a number other than 2, 8, 3, 7. Hence we put the number as it is.We get our one’s place digit as 1.
Now to get tenth place digit
Step 2: We need to strike the last 3 digits of the given number.
In this example the last 3 digits are 791 which we will strike off as shown below
29791
Step 3: We need to find the nearest cube to the remaining number(29).We find that 27 is the nearest cube to 29.
Step 4: We need to find the cube root of the nearest cube(216)
∛27=3
From this step we get our ten’s place digit as 3.
From step 1 and step 4 we get the
∛29791=31
Ans:31
Example 4:Find the cube root of 91125 in 5 seconds.
∛91125=?
In this example, I am going to solve the question step by step.
Step 1:First we need to find the cube root of the last digit of the given number. Here the last digit is 5.5 is a number other than 2, 8, 3, 7. Hence we put the number as it is.We get our one’s place digit as5.
Now to get tenth place digit
Step 2: We need to strike the last 3 digits of the given number.
In this example the last 3 digits are 125 which we will strike off as shown below
91125
Step 3: We need to find the nearest cube to the remaining number(91).We find that 64 is the nearest cube to 91.
Step 4: We need to find the cube root of the nearest cube(216)
∛64=4
From this step we get our ten’s place digit as 4.
From step 1 and step 4, we get the
∛91125=45
Ans: Cube root of 91125 is 45