Square Root

The square root of a number is that number, the product of which itself gives the given number, ie, the square
 root of 400 is 20, the square root of 625 is 25. The square root of a number is denoted by the symbol √ 
 called the radical sign. The expression "√9" is read as "root nine", "radical nine" or "the square root
 of nine".

To Find Square Root of an Integer

(i) By the method of prime factors When a given number is a perfect square, we resolve it into prime factors
 and the product of prime factors, choosing one out of every two.
Example 1 √4356 = ?
        Solution.

4356 = 22 × 32 × 112
√4356 = 2 × 3 × 11 = 66

(ii) By the method of long division This method can be used when the number is large and the factors cannot be
determined easily.
Example 2 √156816 = ?
Solution. 
Explanation Firstly, mark off the digits in pairs starting from the unit's digit. Each pair is called a period.
  Now, 32 = 9 and 42 = 16, So we take 32 = 9 and on subtracting 9 from 15 we 
  get 6 as remainder.
  
            
Now, bring down the next period, ie, 68.
Now, double the root figure already found which is 3 and write it to the left.
Now, from trial and error we find 69 × 9 = 621, which is closest and least to 668.
So, place 9 to the right of 6 changing it to 69. We also put another 9 to the right of the quotient 3 making it 39.
Now, we subtract 621 from 668. we get a remainder 47.
Now, repeat the whole process till there is no period left over to be brought down.
Now, repeat the whole process till there is no period left over to be brought down.
So,      √156816 = 396

To Find the Square Root of a Decimal

Example 3 √1.8225 = ?
Solution. Method 1 
       
√1.8225 = 1.35

Method 2 : √1.8225 = √(18225/10000)
                         = √18225/√10000 = 135/100 = 1.35

To Find the Square Root of a Fraction

Example 4 : √1(13/36) = ?
Solution. √1(13/36) = √(49/36) = √49/√36 = 7/6 = 1(1/6)

1. The square of a number other than unity is either a multiple of 4 or exceeds a multiple of 4 by 1.
2. A perfect square can never end with an odd number of zeroes and 2, 3, 7 and 8 in units place.
3. The square root of an integer is not always an integer, ie, √3, √5, √4 are not integers.
4. √ab = √a × √b
5. √(a/b) = √a/√b
6. √a + √b ≠ √(a + b)
7. √a - √b ≠ √(a - b)

Example 5 : √0.0016 = ?
Solution : √0.0016 = √(16/10000) = √16/√10000 = 4/100 = 0.04 

Example 6 : √125.44 = ?
Solution : Method 1 
        
√125.44 = 11.2 
Method 2 : √125.44 = √(12544/100) = √12544/√100 = 112/10 = 11.2 

Cube Root

The cube root of a number is that number the cube of which itself gives the given number, ie, the cube root of 64 is 4.
The cube root of a number is denoted bythe symbol 3√. The expression 3√8 is read as "cube eight",
or the "cube root of eight".

To Find the Cube Root of an Integer

By the method of prime factors. When a given number is a perfect cube we resolve it into prime factors and take the 
product of prime factors, choosing one out of every three.
Example 7 : 
Solution :
               

So, 3375 = 33 × 53
3√3375 = 3 × 5 = 15 

To find the Cube Root of a Decimal

Example 8 : 3√19.683 = ?
Solution : 3√19.683 = 3√(19683/1000) = 3√39/3√103 = 33/10 = 27/10 = 2.7

To find the Cube Root of a Fraction

Example 9 : 3√1(61/64) = ?
Solution : 3√1(61/64) = 3√(125/64) = 3√125/3√64 = 3√53/3√43 = 5/4 = 1(1/4)
1. 3√ab = 3√a × 3√b
2. 3√(a/b) = 3√a/3√b
3. 3√a + 3√b ≠ 3√(a + b)
4. 3√a - 3√b ≠ 3√(a - b)

Solved Examples

Type 1 To Find the Exact Value

Example 10 : 3√74088 = ?
Solution : 3√74088 = 3√74088 = 3√23 × 73 = 2 × 3 × 7 = 42 
                              
Example 11 : 
                       
Example 12 : (√(625/784) - √(16/49)) ÷ √(81/144) = ?
Solution : (√(625/784) - √(16/49)) = (√625/√784) - (√16/√49) = 25/28 - 4/7 = 9/28
           √81/144 = √81/√144 = 9/12 = 3/4      Hence, ? = 9/28 ÷ 3/4 = 9/28 × 4/3 = 3/7 

Example 13 : 3√√729 = ?
Solution  : √729 = √27 × 27 = 27      Hence, ? = 3√27 = 3√3 × 3 × 3 = 3 

Type 2 To Find the Approximate Value

Example 14 : √2020 = ? 
Solution : 
                 

So, √2020 = 449

Example 15 : 3√12160 × √3140 ÷ 2 = ?
Solution : 3√12160 ≈ 3√23 × 23 × 23 ≈ 23 
           √3140 ≈ √56 × 56 ≈ 56
                   => ? = (23 × 56)/2 = 644