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SQUARE ROOTS 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 .
answer will be 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
Answer will be 2116.
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 ,y1/2=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 √6084 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
6084=2^2*3^2*13^2 3 1521
3 507
13 169
13
√6084=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)86.49 1/2+ (5+?^2) 1/2=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+x^2)1/2=12.3
86.491/2 + (5+x^2 )1/2=12.3
9.3+ (5+x^2 )1/2=12.3
=> (5+x^2 ) 1/2=12.3-9.3
=> (5+x^2 )1/2=3
Squaring on both sides we get
(5+x^2 )=9
x^2 =9-5
x^2 =4
x=2.
Answer is 2.
3.0.000047611/2 equals:
Solution: (4761/10^8) 1/2
47611/2/ 10^81/2 .
69/10000
0.0069.
Answer is 0.0069
4.If 182251/2=135,then the value of
182.251/2 + 1.82251/2 + 0.0182251/2 + 0.000182251/2.
Solution: (18225/100) 1/2+(18225/10000)1/2 +(18225/1000000) 1/2+(18225/100000000)1/2
(18225)/101/2 + (18225)/1001/2 + (18225)/1000 1/2+ (18225)/100001/2
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
=>x^2= (128*162)1/2
=> x^2= (64*2*18*9)1/2
=>x^2= (82*62*32)1/2
=>x^2=8*6*3
=>x^2=144
=>x=12.
6.If 0.13 / p^2=13 then p equals
Solution: 0.13/p^2=13
=>p^2=0.13/13
=1/100
p^2=(1/100)1/2
=>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 (x-1)(y+2)1/2=7 x& y being positive whole numbers then values of x& y are?
Solution: (x-1)(y+2)1/2=7
Squaring on both sides we get
(x-1)(y+2)=7^2
x-1=7 and y+2=7
therefore x=8 , y=5.
Answer x=8 ,y=5.
9.If 3*51/2+121/25=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*5)1/2+6*51/2
=4 51/2+6 1/25=10 51/2
10*2.235
=22.35
Answer is 22.35
10.If 3a=4b=6c and a+b+c=27*291/2 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=(a^2+b^2)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.