SQUARE ROOT & CUBE ROOTS

Square Root & Cube Root

 

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

If number ends in	Unit’s place digit of the square root
1	1 or 9(10-1)
4	2 or 8(10-2)
9	3 or 7(10-3)
6	4or 6(10-4)
5	5
0	0

 

Lets see the logic behind this 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 1² and 9² is 1.

Unit’s place digit of both 2² and 8² is 4

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 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 firstpair 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

 

 

Example 1: √784=?

Step 1: We start by grouping the numbers in pairs of two from right as follows

7 84

 

Step 2: To get the ten’s place digit,

We find that nearest square to 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,

2×3=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

 

 

 

Cube roots of perfect cubes

It may take two-three minutes to find out cube root of a perfect cube by using conventional method. However we can find out cube roots of perfect cubes very fast, say in one-two seconds using Vedic Mathematics.

We need to remember some interesting properties of numbers to do these quick mental calculations which are given below.

 

Points to remember  for speedy  calculation of cube roots

1. To calculate cube root of any perfect cube quickly, we need to remember the cubes of 1 to 10 which is given below.

13	=	1
23	=	8
33	=	27
43	=	64
53	=	125
63	=	216
73	=	343
83	=	512
93	=	729
103	=	1000

2. From the above cubes of 1 to 10, we need to remember an interesting property.

13 = 1	=>	If last digit of the perfect cube = 1, last digit of the cube root = 1
23 = 8	=>	If last digit of the perfect cube = 8, last digit of the cube root = 2
33 = 27	=>	If last digit of the perfect cube = 7, last digit of the cube root = 3
43 = 64	=>	If last digit of the perfect cube = 4, last digit of the cube root = 4
53 = 125	=>	If last digit of the perfect cube =5, last digit of the cube root = 5
63 = 216	=>	If last digit of the perfect cube = 6, last digit of the cube root = 6
73 = 343	=>	If last digit of the perfect cube = 3, last digit of the cube root = 7
83 = 512	=>	If last digit of the perfect cube = 2, last digit of the cube root = 8
93 = 729	=>	If last digit of the perfect cube = 9, last digit of the cube root = 9
103 = 1000	=>	If last digit of the perfect cube = 0, last digit of the cube root = 0

 

It’s very easy to remember the relations given above because

1	->	1	(Same numbers)
8	->	2	(10’s complement of 8 is 2 and 8+2 = 10)
7	->	3	(10’s complement of 7 is 3 and 7+3 = 10)
4	->	4	(Same numbers)
5	->	5	(Same numbers)
6	->	6	(Same numbers)
3	->	7	(10’s complement of 3 is 7 and 3+7 = 10)
2	->	8	(10’s complement of 2 is 8 and 2+8 = 10)
9	->	9	(Same numbers)
0	->	0	(Same numbers)

 

Also see
8 ->  2 and 2 ->  8
7 -> 3 and 3-> 7

 

 

 

 

 

Questions

Level-I

1.	The cube root of .000216 is:
	A. .6 B. .06 C. 77 D. 87

 

2.	What should come in place of both x in the equation x = 162 . 128 x
	A. 12 B. 14 C. 144 D. 196

 

3.	The least perfect square, which is divisible by each of 21, 36 and 66 is:
	A. 213444 B. 214344 C. 214434 D. 231444

 

4.	1.5625 = ?
	A. 1.05 B. 1.25 C. 1.45 D. 1.55

 

5.	If 35 + 125 = 17.88, then what will be the value of 80 + 65 ?
	A. 13.41 B. 20.46 C. 21.66 D. 22.35
6.	If a = 0.1039, then the value of 4a2 – 4a + 1 + 3a is:
	A. 0.1039 B. 0.2078 C. 1.1039 D. 2.1039

 

7.	If x = 3 + 1 and y = 3 – 1 , then the value of (x2 + y2) is: 3 – 1 3 + 1
	A. 10 B. 13 C. 14 D. 15

 

8.	A group of students decided to collect as many paise from each member of group as is the number of members. If the total collection amounts to Rs. 59.29, the number of the member is the group is:
	A. 57 B. 67 C. 77 D. 87

 

9.	The square root of (7 + 35) (7 – 35) is
	A. 5 B. 2 C. 4 D. 35

 

 

 

10.	If 5 = 2.236, then the value of 5 – 10 + 125 is equal to: 2 5
	A. 5.59 B. 7.826 C. 8.944 D. 10.062       Level-II
11.	625 x 14 x 11 is equal to: 11 25 196
	A. 5 B. 6 C. 8 D. 11

 

12.	0.0169 x ? = 1.3
	A. 10 B. 100 C. 1000 D. None of these

 

13.	3 – 1 2 simplifies to: 3
	A. 3 4 B. 4 3 C. 4 3 D. None of these

 

14.	How many two-digit numbers satisfy this property.: The last digit (unit’s digit) of the square of the two-digit number is 8 ?
	A. 1 B. 2 C. 3 D. None of these

 

15.	The square root of 64009 is:
	A. 253 B. 347 C. 363 D. 803

 

 

16. √29929 = ?

A. 173 B. 163 C. 196 D. 186

 

 

 

 

 

 

17. √106.09 = ?
A.	10.6
B.	10.5
C.	10.3
D.	10.2

 

18.  ?/√196 = 5
A.	76
B.	72
C.	70
D.	75

 

Answers

Level-I

 

Answer:1 Option B

 

Explanation:

(.000216)1/3	=		216		1/3
			106

 

=		6 x 6 x 6		1/3
		102 x 102 x 102

 

=	6
	102

 

=	6
	100

= 0.06

 

Answer:2 Option A

 

Explanation:

Let	x	=	162
	128		x

Then x² = 128 x 162

= 64 x 2 x 18 x 9

= 8² x 6² x 3²

= 8 x 6 x 3

= 144.

x = 144 = 12.

 

Answer:3 Option A

 

Explanation:

L.C.M. of 21, 36, 66 = 2772.

Now, 2772 = 2 x 2 x 3 x 3 x 7 x 11

To make it a perfect square, it must be multiplied by 7 x 11.

So, required number = 2² x 3² x 7² x 11² = 213444

 

Answer:4 Option B

 

Explanation:

1|1.5625( 1.25

|1

|——-

22| 56

| 44

|——-

245| 1225

| 1225

|——-

|    X

|——-

1.5625 = 1.25.

 

 

Answer:5 Option D

 

Explanation:

35 + 125 = 17.88

35 + 25 x 5 = 17.88

35 + 55 = 17.88

85 = 17.88

5 = 2.235

80 + 65 = 16 x 5 + 65

= 45 + 65

= 105 = (10 x 2.235) = 22.35

 

 

 

Answer:6 Option C

 

Explanation:

4a² – 4a + 1 + 3a = (1)² + (2a)² – 2 x 1 x 2a + 3a

= (1 – 2a)² + 3a

= (1 – 2a) + 3a

= (1 + a)

= (1 + 0.1039)

= 1.1039

 

Answer:7 Option C

 

Explanation:

x =	(3 + 1)	x	(3 + 1)	=	(3 + 1)2	=	3 + 1 + 23	= 2 + 3.
	(3 – 1)		(3 + 1)		(3 – 1)		2

 

y =	(3 – 1)	x	(3 – 1)	=	(3 – 1)2	=	3 + 1 – 23	= 2 – 3.
	(3 + 1)		(3 – 1)		(3 – 1)		2

x² + y² = (2 + 3)² + (2 – 3)²

= 2(4 + 3)

= 14

 

Answer:8 Option C

 

Explanation:

Money collected = (59.29 x 100) paise = 5929 paise.

Number of members = 5929 = 77

 

 

Answer:9 Option B

 

Explanation:

(7 + 35)(7 – 35)

=

(7)2 – (35)2

= 49 – 45  = 4  = 2

 

 

Answer:10 Option B

 

Explanation:

5	–	10	+ 125	=	(5)2 – 20 + 25 x 55
2		5			25

 

=	5 – 20 + 50
	25

 

=	35	x	5
	25		5

 

=	355
	10

 

=	7 x 2.236
	2

 

= 7 x 1.118

 

= 7.826

 

 

Level-II

Answer:11 Option A

 

Explanation:

Given Expression =	25	x	14	x	11	= 5.
	11		5		14

 

 

 

Answer:12 Option B

 

Explanation:

Let 0.0169 x x = 1.3.

Then, 0.0169x = (1.3)² = 1.69

x =	1.69	= 100
	0.0169

 

 

 

Answer:13 Option C

 

Explanation:

	3 –	1		2	= (3)2 +		1		2	– 2 x 3 x	1
		3					3				3

 

= 3 +	1	– 2
	3

 

= 1 +	1
	3

 

=	4
	3

 

 

 

Answer:14 Option D

 

Explanation:

A number ending in 8 can never be a perfect square.

 

 

Answer:15 Option A

 

Explanation:

2 |64009( 253      |4      |———-45  |240      |225      |———-503| 1509      |  1509      |———-      |     X      |———-

64009 = 253.

 

 

Answer:16 Option A

 

Explanation:
√29929 = So, √29929 = 173

 

 

Answer:17 Option C

 

Answer:18 Option C

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