SQUARES AND CUBES With ROOTS

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SQUARES AND CUBES With ROOTS

SQUARES AND CUBES With ROOTS

SQUARES AND SQUARE ROOTS

When a number is multiplied by same number, the result is called square of the number.
X x X = x2
x2 is the square of the number x.
EASY SQUARING METHODS

TYPE-I

Square number ending in 5
Step I

Multiply the first digit by itself plus one.
Eg:- 35^2
So 3 x (3 1) = 12
Step II

Write the number 5^2 = 25
next to the result from Step I
So 1225
.'. 35^2 = 1225
EXAMPLES

1.65^2
1st=>6*(61)=42
2nd =>5^2=25
.'.65^2=4225
2.95^2
1st=>9*(91)=90
2nd=>5^2=25
.'.95^2=9025
SQUARES OF 1ST 30 NATURAL NUMBERS

1^2 =1 16^2= 256
2^2 = 4 17^2 =289
3^2 =9 18^2= 324
4^2 = 16 19^2=361
5^2 = 25 20^2=400
6^2 =36 21^2= 441
7^2 =49 22^2=484
8^ = 64 23^2= 529
9^2 = 81 24^2= 576
10^2 =100 25^2=625
11^2 = 121 26^2=676
12^2 144 : 27^2=729'
13^2 =169 28^2= 784
14^2 = 196 29^2 =841
15^^2 =225 30^2 =900
TYPE - II

Square numbers starting in 5
Step I
Add 25 to the ones digit Eg:- 532
25 3 = 28
Step II

Square the ones digit number. If the result is a single digit, put a 0 in front of it.
3^2 = 09 So 532= 2809
EXAMPLES

156^2
1st => 25 6 = 31
2nd => 62 = 36
So 562 = 3136
2.51^2
1st => 25 1 = 26
2nd=>1^2=01
So 512 = 2601
TYPE - III

Square numbers between 90 and 28
Step I
Subtract the number from 100
Eg:- 97^2
100-97 = 3
Step II

Subtract this number from original number
ie,97-3=94
Step III

Square the result from step I.If the result is a single digit,put a 0 in front of it.
ie,3^2 =09
So 97^2=9409
EXAMPLES

1.92^2
1st=>100- 92=8
2nd=>92 - 8=84
3rd=>8^2=64
So 92^2 =8464
2.94^2
1st=>100-94=6
2nd=>94-6=88
3rd=>6^2 = 36
So 94^2=36
3.99^2
1st=>100-99=1
2nd=>99-1=98
3rd=>1^2=01
So 99^2=9801
TYPE-IV

Square number between 40 and 49
Step I
Subtract the number from 50.
Eg:- 48^2
50 - 48=2
Step II

Substract the result (from step I) from 25.
ie,25-2=23
Step III

square the result from step I.If the result is a single digit put a 0 infront of it.
ie,2^2=04
So 48^2=2304
EXAMPLES

1.42^2
1st =>50 - 42=8
2nd=>25 - 8=17
3rd=>8^2=64
So 42^2=1764

CUBES AND CUBE ROOTS

*This is an interesting topic in Mathematics where you can solve problems like playing a game. Here, we will show you how to find cube roots through a shortcut. This will help you in saving your precious time.

CUBES

A number when multiplied by itself 3 times gives the cube of that number.
Eg:- 3^3 = 3x3x3 = 27
CUBES
1^3 = 1
2^3 = 8
3^3=27
4^3 = 64
5^3 = 125
63 = 216
73 = 343
8^=5212
9^3=729
10^3=1000
11^3 = 1331
12^3 = 1728
13^3 = 2197
14^3 = 2744
15^3 = 2375
16^3 = 4096
17^3 = 4913
18^3 = 5832
19^3 = 6859
20^3 = 8000

EXERCISES

1.If 6^3 7^2 x = 9^3 2^3 - 1^3, find the value of x
6^3 7^2 x = 9^3 2^3 - 1^3
216 49 x = 729 8-1
265 x = 736
.'. x = 471
2.If the surface area of a Cube is 216m^2, find the volume of the cube.
Surface area = 216
ie, 6a2 = 216
a2 = 36
.'.a=6
Volume of the cube = a3 = 6x6x6 = 216m3
3.(3^3 - 2^3) (6^3 - 5^3)= x/2 Find the value of x.
(27 - 8) (216 - 125)=x/2
19 91=x/2
110=x/2
.'.x=220

CUBEROOT

*The cube root of a number is one of three equal factors which when multiplied gives that number.
Eg:- 3 64 =4
We will follow these tricks to find the cube root easily.

Steap:

Ignore the last 3 digits of the number. Consider the remaining numbers as one.
Eg:- 21952
21, 952 (ignoring last 3 digits)
consider 21

POINTS TO REMEMBER

The numbers which are both squares as well as cubes.
*1^2=1,1^3=1
*8^2 64,4^3 = 64
*27^2 = 729,
*9^3 =729
2.44^2
1st =>50 - 44=6
2nd=>25- 6=19
3rd=>6^2=36
So 44^2=1936
3.41^2
1st =>50 - 41=9
2nd=>25 - 9=16
3rd=>9^2=81
So 41^2=1681

POINTS TO REMEMBER

Properties of a Perfect square
*No perfect square ends with 2,3,7,8
*No perfect square ends with an odd number of zeros
*The square of a number other than unity is either a multiple of 4 or exceeds a multiple of 4 by 1.

SQUARE ROOT

General method to find the square root
Mark off the digits in pairs from right and then find the square root
Eg:- 219961
4 21,99,61
16
366 599
516
929 8361
8361
0
.'./219961=469

POINTS TO REMEMBER

*If a and b are perfect squares,
then /a/b=/a/ /b
*If b is a perfect
Square,/a/b
=/ab//bb=/ab/b

Step II:

From the table of cubes, check which number's cube is less than or equal to the result from step I.
ie, 23 = 8 < 21
Then 2 will be the left part of our answer.

Step III:

If the last digit of the question is 7, then our answer's right part will be 8. (because 3^3 = 27).
Similarly, here the last digit is 2. So the right part will be 8.
So 3/21952 = 28

EXAMPLES

1.
3/46656
Step I : 46,656.
Step II : 3^3 = 27 < 46
So left part is 3.
Step III : Last digit of 46656 is 6.
So right part is 6 (6^3 is 216)
So answer is 36.
2.
3/185193
Step I: 185, 193
Step II: 5^3 = 125 <185
So left part is 5
Step III: Last digit of 185193 is 3
So right part is 7 (7^3 = 343)
So answer is 57
3.
3/704969
Step I: 704,969
Step II: 8^3 = 512 < 704
So left part is 8.
Step III: Last digit of 704, 969 is 9.
So right part is 9 (9^3 = 729)
So answer is 89.
4.
3/405224
Step I: 405, 224
Step II: 7^3 = 343 < 405
So left part is 7
Step III: Last digit of 405224 is 4.
So right part is 4 (4^3 = 64)
So answer is 74.
5.
3/27 6^3 x = 9^3 3/8 . Find x.
3 216 x = 729 2
x = 731-219
x = 512
6.
If the cube of a number x is the same as the square of another number y and y = 2x, find the product of those numbers.
We know 3 numbers which are both cubes and squares. They are 1,64 and 729.
Given, y = 2x
So it will be 64
ie, 4^3 = 64
8^2 =64
.'.Numbers are 4 and 8 and their product is 32.

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