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
-------------------------
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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 ,√y=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 + √(5+x2)=12.3
solutin:86.491/2 + (5+x1/2 )=12.3
9.3+ √(5+x1/2 )=12.3
=> √(5+x1/2 ) =12.3-9.3
=> √(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.√ 0.00004761 equals:
Solution: √ (4761/108)
√4761/√ 108
. 69/10000
0.0069.
Answer is 0.0069
4.If √18225=135,then the value of
√182.25 + √1.8225 + √ 0.018225 + √0.00018225.
Solution: √(18225/100) +√(18225/10000) +
√(18225/1000000) +√(18225/100000000)
=√(18225)/10 + (18225)1/2/100 +
√(18225)/1000 + √(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=√(1/100)
=>p=1/10
therefore p=0.1
Answer is 0.1
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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 √(x-1)(y+2)=7 x& y being positive whole numbers then
values of x& y are?
Solution: √(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*√29 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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