The square root of a number has the exponent of 1/2. The square root formula is used to find the square root of a number. We know the exponent formula:
When n = 2, we call it square root. We can use any of the above methods for finding the square root, such as prime factorization, and so on. 9^{1/2} = √9 = √(3×3) = 3. So, the formula for writing the square root of a number is √x= x^{1/2}
The cube root formula helps in the calculation of the cube root of any given number that is expressed in a radical form using the symbol ∛. It can be calculated by first finding out the prime factorization of the given number and then later applying the cube root formula. Suppose, x is any number such that, x = y × y × y.
Important Points:
(a)
(b)
We can estimate the square root of a perfect square number using a trick. To determine the square root without using a long division, we must know the unit digits of squares of the first ten numbers.
1 = 1
2 = 4
3 = 9
4 = 6
5 = 5
6 = 6
7 = 9
8 = 4
9 = 1
10 = 0
It is possible to calculate the square root of a number having 4 digits as well as 5 digits. Let us learn to calculate the square root of numbers using the trick.
Steps to find units digit if the unit digit obtained in Step 2 is apart from 5 or 10 are:
The steps to estimate the square root of a 5 digit number are:
Steps to find units digit if the unit digit obtained in Step 2 is apart from 5 or 10 are:
Example: Find the square root of 10816.
Solution: The steps to determine the square root of 10816 are:
First we need to remember cubes of 1 to 10 and unit digits of these cubes. The figure below shows the unit digits of cubes (on the right) of numbers from 1 to 10 (on the left).
1 = 1
2 = 8
3 = 7
4 = 4
5 = 5
6 = 6
7 = 3
8 = 2
9 = 9
10 = 0
Now with reference to above we can definitely say that:
Whenever unit digit of a number is 9, the unit digit of the cube of that number will also be 9. Similarly, if the unit digit of a number is 9, the unit digit of the cube root of that number will also be 9. Similarly, if unit digit of a number is 2, unit digit of the cube of that number will be 8 and vice versa if unit digit of a number is 8, unit digit of the cube root of that number will be 2. Similarly, it will be applied to unit digits of other numbers as well.
Let’s see this with the help of an example. Note that this method works only if the number given is a perfect cube.
Find the cube root of 474552.
Let us take another example.
Find the cube root of 250047.
Time saving techniques are paramount when we have to deal with Quant questions in any competitive exams, which is missing a place in the provided material.
Many times we need to find a square of a number and it gets difficult to remember it beyond 30. So, here is a trick…..
Suppose, we need to find the square of 47.
Step 1: If the number is between 25 and 50.
Find out by how much the given number is smaller than 47. In the above case, it is 3.
Step 2: Write the square of this number in unit’s and ten’s place in this manner Square of 3 is 9.
It is a single digit number, so we can write it as 09
Step 3: Find the difference between the given number and 25
4725 = 22
Therefore, square of 47 will be 2209.
This is true for the square of any number between 25 and 50.
Example: Square of 73
Step 1: find the difference between the number (73) and 50, which is 23.
Step 2: Find the square of the difference, 23^{2} = 529. Keep the last two number aside which will be the last two digits of the square of 73.
Step 3: Find the difference between the number for which we have to find the square and add the difference with 5(which is the first digit of the square of the difference)
⇨ (73  25) + 5 = 53. This number will be the first two digits of the square.
The square of the number 73 will be = 5329
Example: Square of 88
Step 1: Subtract the number from 100.. (100  88 = 12)
Step 2: Find the square of the number obtained .. 12^{2 }= 144
Last two digits of the square of this number will be last 2 digits of square of 88.
Step 3: The first two digits will be obtained by adding the first digit of square of 12. i.e. 1 and the difference between 88 and 12 (88  12 = 76)
76 + 1 = 77
The square of the number 88 will be = 7744
Square of 87 = (87  13)…….13^{2}
74…………….69
7569
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1. How can I find the square root of a number easily? 
2. Is there a shortcut to finding cube roots quickly? 
3. Can square roots and cube roots be negative numbers? 
4. How can I estimate square roots and cube roots without a calculator? 
5. Are there any tricks to mentally calculate square roots and cube roots? 

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