### APTITUDE – problems on SQUARE AND CUBE ROOTS

**Formula:**

The Product of two same numbers in easiest way as follow.

Example:let us calculate the product of 96*96

Solution: Here every number must be compare with the 100.

See here the given number 96 which is 4 difference with the 100.

so subtract 4 from the 96 we get 92 ,then the square of the

number 4 it is 16 place the 16 beside the 92 we get answer

as 9216.

9 6

– 4

————–

9 2

————–

4*4=16

9 2 1 6

therefore square of the two numbers 96*96=9216.

Example: Calculate product for 98*98

Solution: Here the number 98 is having 2 difference when compare

to 100 subtract 2 from the number then we get 96 square the

number 2 it is 4 now place beside the 96 as 9604

9 8

– 2

————-

9 6

————-

2*2=4

9 6 0 4.

so, we get the product of 98*98=9604.

Example: Calculate product for 88*88

Solution: Here the number 88 is having 12 difference when compare

to 100 subtract 12 from the 88 then we get 76 the square of the

number 12 is 144 (which is three digit number but should place

only two digit beside the 76) therefore in such case add one to

6 then it becomes 77 now place 44 beside the number 77 we will get

7744.

88

-12

————

76

———–

12*12=144

76

+ 144

——————–

7744

——————–

Example: Find the product of the numbers 46 *46?

Solution:consider the number 50=100/2. Now again go comparision with

the number which gets when division with 100.here consider the number

50 which is nearer to the number given. 46 when compared with the

number 50 we get the difference of 4. Now subtract the number 4 from

the 46, we get 42. As 50 got when 100 get divided by 2.

so, divided the number by 2 after subtraction.

42/2=21 now square the the number 4 i.e, 4*4=16

just place the number 16 beside the number 21

we get 2116.

4 6

4

—————-

4 2 as 50 = 100/2

42/2=21

now place 4*4=16 beside 21

2 1 1 6

Example: Find the product of the numbers 37*37

Solution:

consider the number 50=100/2

now again go comparision with the number which gets when

division with 100.

here consider the number 50 which is nearer to the number given.

37 when compared with the number 50 we get the difference of 13.

now subtract the number 13 from the 37, we get 24.

as 50 got when 100 get divided by 2.

so, divided the number by 2 after subtraction.

24/2=12

now square the the number 13 i.e, 13*13=169

just place the number 169 beside the number 21

now as 169 is three digit number then add 1 to 2 we get

1t as 13 then place 69 beside the 13

we get 1369.

3 7

1 3

—————–

2 4 as 50 = 100/2

24/2=12

square 13* 13=169

1 2

+ 1 6 9

———————–

1 3 6 9

————————-

Example: Find the product of 106*106

Solution: now compare it with 100 ,

The given number is more then 100

then add the extra number to the given number.

That is 106+6=112

then square the number 6 that is 6*6=36

just place beside the number 36 beside the 112,then

we get 11236.

1 0 6

+ 6

———————

1 1 2

——————–

now 6* 6=36 place this beside the number 112, we get

1 1 2 3 6

Square root: If x2=y ,we say that the square root of y

is x and we write ,âˆšy=x.

Cube root: The cube root of a given number x is the number

whose cube is x. we denote the cube root of x by x1/3 .

**Examples:**

**1.Evaluate 60841/2 by factorization method.**

Solution: Express the given number as the product of prime

factors. Now, take the product of these prime factors choosing

one out of every pair of the same primes. This product gives the

square root of the given number.

Thus resolving 6084 in the prime factors ,we get 6084

2 6024

2 3042

3 1521

3 507

13 169

13

6084=21/2 *31/2 *131/2

60841/2=2*3*13=78.

Answer is 78.

**2.what will come in place of question mark in each of the following
questions?**

i)(32.4/?)1/2 = 2

ii)86.491/2 + (5+?1/2)2 =12.3

Solution: 1) (32.4/x)1/2=2

Squaring on both sides we get

32.4/x=4

=>4x=32.4

=>x=8.1

Answer is 8.1

ii)86.491/2 + âˆš(5+x2)=12.3

solutin:86.491/2 + (5+x1/2 )=12.3

9.3+ âˆš(5+x1/2 )=12.3

=> âˆš(5+x1/2 ) =12.3-9.3

=> âˆš(5+x1/2 )=3

Squaring on both sides we get

(5+x1/2 )=9

x1/2 =9-5

x1/2 =4

x=2.

Answer is 2.

**3.âˆš 0.00004761 equals:**

Solution: âˆš (4761/108)

âˆš4761/âˆš 108

. 69/10000

0.0069.

Answer is 0.0069

**4.If âˆš18225=135,then the value of **

âˆš182.25 + âˆš1.8225 + âˆš 0.018225 + âˆš0.00018225.

Solution: âˆš(18225/100) +âˆš(18225/10000) +

âˆš(18225/1000000) +âˆš(18225/100000000)

=âˆš(18225)/10 + (18225)1/2/100 +

âˆš(18225)/1000 + âˆš(18225)/10000

=135/10 + 135/100 + 135/1000 + 135/10000

=13.5+1.35+0.135+0.0135=14.9985.

Answer is 14.9985.

**5.what should come in place of both the question
marks in the equation (?/ 1281/2= (162)1/2/?) ?**

Solution: x/ 1281/2= (162)1/2/x

=>x1/2= (128*162)1/2

=> x1/2= (64*2*18*9)1/2

=>x2= (82*62*32)

=>x2=8*6*3

=>x2=144

=>x=12.

**6.If 0.13 / p1/2=13 then p equals**

Solution: 0.13/p2=13

=>p2=0.13/13

=1/100

p2=âˆš(1/100)

=>p=1/10

therefore p=0.1

Answer is 0.1

**7.If 13691/2+(0.0615+x)1/2=37.25 then x is equals to:**

Solution

37+(0.0615+x)1/2=37.25(since 37*37=1369)

=>(0.0615+x)1/2=0.25

Squaring on both sides

(0.0615+x)=0.0625

x=0.001

x=10-3.

Answer is 10-3.

**8.If âˆš(x-1)(y+2)=7 x& y being positive whole numbers then
values of x& y are?**

Solution: âˆš(x-1)(y+2)=7

Squaring on both sides we get

(x-1)(y+2)=72

x-1=7 and y+2=7

therefore x=8 , y=5.

Answer x=8 ,y=5.

**9.If 3*51/2+1251/2=17.88.then what will be the
value of 801/2+6*51/2?
**

Solution: 3*51/2+1251/2=17.88

3*51/2+(25*5)1/2=17.88

3*51/2+5*51/2=17.88

8*51/2=17.88

51/2=2.235

therefore 801/2+6 51/2=(16*51/2)+6*1/25

=4 51/2+6 51/2

=10*2.235

=22.35

Answer is 22.35

**10.If 3a=4b=6c and a+b+c=27*âˆš29 then Find c value is:**

Solution: 4b=6c

=>b=3/2*c

3a=4b

=>a=4/3b

=>a=4/3(3/2c)=2c

therefore a+b+c=27*291/2

2c+3/2c+c=27*291/2

=>4c+3c+2c/2=27*291/2

=>9/2c=27*291/2

c=27*291/2*2/9

c=6*291/2

**11.If 2*3=131/2 and 3*4=5 then value of 5*12 is**

Solution:

Here a*b=(a2+b2)1/2

therefore 5*12=(52+122)1/2

=(25+144)1/2

=1691/2

=13

Answer is 13.

**12.The smallest number added to 680621 to make
the sum a perfect square is**

Solution: Find the square root number which

is nearest to this number

8 680621 824

64

162 406

324

1644 8221

6576

1645

therefore 824 is the number ,to get the nearest

square root number take (825*825)-680621

therefore 680625-680621=4

hence 4 is the number added to 680621 to make it

perfect square.

**13.The greatest four digit perfect square number is **

Solution: The greatest four digit number is 9999.

now find the square root of 9999.

9 9999 99

81

189 1819

1701

198

therefore 9999-198=9801 which is required number.

Answer is 9801.

**14.A man plants 15376 apples trees in his garden and arranges
them so, that there are as many rows as there are apples trees
in each row .The number of rows is.**

Solution: Here find the square root of 15376.

1 15376 124

1

22 53

44

244 976

976

0

therefore the number of rows are 124.

**15.A group of students decided to collect as many paise from
each member of the group as is the number of members. If the
total collection amounts to Rs 59.29.The number of members
in the group is:**

Solution: Here convert Money into paise.

59.29*100=5929 paise.

To know the number of member ,calculate the square root of 5929.

7 5929 77

49

147 1029

1029

0

Therefore number of members are 77.

**16.A general wishes to draw up his 36581 soldiers in the form
of a solid square ,after arranging them ,he found that some of
them are left over .How many are left?**

Solution: Here he asked about the left man ,So find the

square root of given number the remainder will be the left man

1 36581 191

1

29 265

261

381 481

381

100(since remaining)

Therefore the left men are 100.

**17.By what least number 4320 be multiplied to obtain number
which is a perfect cube?**

Solution: find l.c.m for 4320.

2 4320

2 2160

2 1080

2 540

2 270

3 135

3 45

3 15

5

4320=25 * 33 * 5

=23 * 33 * 22 *5

so make it a perfect cube ,it should be multiplied by 2*5*5=50

Answer is 50.

**18.3(4*12/125)1/2=?**

Solution: 3(512/125)1/2

3(8*8*8)1/2/(5*5*5)

3(83)1/2/(53)

((83)/(53))1/3

=>8/5 or 1 3/5.

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