Square root and cube root

Square

A number whose power is 2. If 'N' is a number, then the square of N is N² which is equal to N × N.
When a number is multiplied by itself, then the result of this multiplication is called the square of the number.
For example- Square of 2 = 2² = 2 × 2 = 4
Square of 4 = 4² = 4 × 4 = 16
Square of 5 = 5² = 5 × 5 = 25

Methods to Find Square :-

By this method we find the square of any number whose last digit is 5.
Step 1 :- Square the unit's digit and write in the end.
Step 2 :- Multiply the ten's digit by the successor of the ten's digit and write it as the first digit.

Example- Find the square of 15.
Solution:-
step 1 :- 5² = 25
step 2 :- 1 × 2 = 2
∴ ( 15 )² = 225

Example- Find the square of 35.
Solution:- step 1 :- 5² = 25
step 2 :- 3 × 4 = 12
∴ ( 35 )² = 1225

To find Square using Identity :-

1. ( a + b )² = a² + 2ab + b²

Ex- Find the square of 103.
Solution:- ( 103 )² = ( 100 + 3 )²
= 100² + 3² + 2 × 100 × 3
= 10000 + 9 + 600
Hence , ( 103 )² = 10609

2. ( a - b )² = a² - 2ab + b²

Ex- Find the square of 998.
Solution:- ( 998 )² = ( 1000 - 2 )
= 1000² + 2² - 2 × 1000 × 2
= 1000000 + 4 - 4000
Hence , ( 998 )² = 996004

Properties of Square

Unit digit after squaring
1² = 1
2² = 4
3² = 9
4² = 6
5² = 5
6² = 6
7² = 9
8² = 4
9² = 1
10² = 0

1. A number whose last digits are 2, 3, 7 or 8 can never be a perfect square.
For example- 1532, 1643 etc.

2. A number whose last digits are 0, 1, 4, 5, 6 or 9 may be or may not be a perfect square.
For example- 100, 1000,121,131 etc.

3. In case a number has last digits as 1, 5 or 6, then on squaring that number, its last digit remains same as before.
For example- 1² = 1, 15² = 225, 16² = 216 etc.

4. If a number's last digits are 1 or 9, then after squaring, the calculated square number ends with 1.
For example- 11² = 121, 19² = 361 etc.

5. If a number whose last digits are 2 or 8, then after squaring, the calculated square number ends with 4.
For example- 12² = 144, 18² = 324 etc.

6. If a number whose last digits are 3 or 7, then after squaring, the calculated square number ends with 9.
For example- 13² = 169, 17² = 289 etc.

7. If a number whose last digits are 4 or 6, then after squaring, the calculated square number ends with 6.
For example- 14² = 196, 16² = 256 etc.

8. The square of any number always ends with 0, 1, 4, 5, 6 or 9.

9. The square of any negative number is always positive.
For example- -7² = 49, -17² = 289 etc.

10. Square of an even number is even.
For example- 16² = 256, 18² = 324 etc.

11. Square of an odd numbers is odd.
For example- 11² = 121, 13² = 169 etc.

Numbers between two consecutive square numbers :-

If two consecutive numbers are n and ( n + 1 ), then 2n numbers lie between the square of n and ( n + 1 ).

Ex- How many natural numbers lie between 7² and 8².
Solution:- Here, n = 7
Natural numbers lie between 7² and 8² = 2n
= 2 × 7
= 14

Sum of Two Consecutive Number :-

Square of any odd number is equal to the sum of two consecutive number.
For example- 3² = 9 = 4 + 5
5² = 25 = 12 + 13
7² = 49 = 24 + 25

Square of n is equal to the sum of	( n² + 1 )	and	( n² - 1 )
	2		2

Example :- Express the sum of two consecutive numbers of 15².
Solution:- Here, n = 15 , n² = 225

∴ Sum =	( n² + 1 )	and	( n² - 1 )
	2		2

Sum =	( 225 + 1 )	and	( 225 - 1 )
	2		2

Sum =	226	and	224	= 113 and 112
	2		2

Sum of first odd natural numbers

1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16
1 + 3 + 5 + 7 + 9 = 25

Sum of first n odd natural numbers is equal to n².
Where, n = number of terms

Example- Find the sum of 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19.
Solution:- Here, n = 10
∴ sum = n²
= 10²
= 100

Example-Find the sum of first 13 odd number.
Solution:- here, n = 13
Sum = n²
= 13²
= 169

Difference between square of two consecutive numbers :-

a² - b² = ( a + b ) ( a - b )
When two consecutive numbers a and b ( a > b ), then
Difference between a² and b² = ( a + b ) ( a - b )
Here,( a - b ) = 1, because difference between two consecutive number is 1.

∴ Difference between square of two consecutive numbers a and b = ( a + b )

Example- Find the difference between 15² and 14².
Solution:- Here, a = 15 and b = 14
∴ 15² - 14² = ( a + b )
= 15 + 14
= 29

Square Root

When square root of a number is multiplied by itself, it gives the number.
Symbol of square root is √

Square roof of a perfect square by prime factorization method.

To find square roof of a number, first find the prime factors of the given number. Then make pairs of similar factors and then take the product of the prime factors choosing only one factor out of every pair.

Ex- Find the square root of 729.

Solution:- By prime factorization, we get
729 = 3 × 3 × 3 × 3 × 3 × 3
√729 = 3 × 3 × 3 = 27

Cube :-

A number whose power is 3. If 'N' is a number, then the cube of N is N³
which is equal to N × N × N.
When a number is multiplied by twice with itself, then the result is called cube of that number.
For example- cube of 2 = 2³ = 2 × 2 × 2 = 8
cube of 3 = 3³ = 3 × 3 × 3 = 27
cube of 4 = 4³ = 4 × 4 × 4 = 64

Method to find cube :-

To find cube using Identity :-

1.( a + b )³ = a³ + b³ + 3ab( a + b )

Example :- Find the cube of 13.
Solution:- ( 13 )³ = ( 10 + 3 )³
= 10³ + 3³ + 3 × 10 × 3 ( 10 + 3 )
= 1000 + 27 + 90 × 13
= 1000 + 27 + 1170
= 2197

2.( a - b )³ = a³ - b³ - 3ab( a - b )

Example :- Find the cube of 97.
Solution:- ( 97 )³ = ( 100 - 3 )³
= 100³ - 3³ - 3 × 100 × 3 ( 100 - 3 )
= 1000000 - 27 - 900 ×97
= 1000000- 27 - 87300
= 1000000 - 87327 = 912673

Properties of Cubes :-

After cube, unit digits are
1³ = 1
2³ = 8
3³ = 7
4³ = 4
5³ = 5
6³ = 6
7³ = 3
8³ = 2
9³ = 9
10³ = 0

1. Cube of a negative number is negative.
For example- ( -2 )³ = -8, ( -3 )³ = -27 etc.

2. A number whose last digits 0, 1, 4, 5, 6 or 9 after cubing ends up with same last digit.
For example- 11³ = 1331, 14³ = 2744, 19³ = 6859 etc.

3. Cube of an even number is even.
For example- 2³ = 8, 4³ = 64, 6³ = 216 etc.

4. Cube of an odd number is odd.
For example- 3³ = 27, 5³ = 125, 7³ = 343 etc.

5. Sum of the cubes of first n natural number is equal to the square of their sum.

Example- Find the sum of 1³ + 2³ + 3³ + 4³.
Solution:- 1³ + 2³ + 3³ + 4³ = ( 1 + 2 + 3 + 4 )²
= 10²
= 100

Cube Root :-

When the cube root of a number is multiplied thrice, it gives the number.
For example 3 × 3 × 3 = 27, then 3 is the cube root of 27.
∴ ³√27 = 3

Method to find cube root :-

Using Prime Factorization Method :-

To find cube root of a number, first find the prime factors of the given number, then make group of 3 of similar factors. Then take the product of the prime factors after choosing only one factor out of every group.

Ex- Find the cube root of 512.
Solution:- By prime factorization, we get
512 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
∴ ³√512 = 2 × 2 × 2 = 8