## SQUARE AND CUBE ROOTS

### Introduction

• The straight line method of multiplying two numbers

1. Suppose you were multiplying two 2 digit numbers like 43 × 78. The first objective would be to get the unit’s digit. In order to do this we just need to multiply the units’ digit of both the numbers. Thus, 3 × 8 would give us 24. Hence, we would write 4 in the units’ digit of the answer and carry over the digit 2 to the tens place.

2. Step 2: Finding the tens’ place digit Multiply the units digit of second number to tens digit of first. Multiply units digit of first to tens digit of second. Add both results and any carry forward received from the operation on step 1. Thus we would get: 8 (units digit of the second number) × 4 (tens digit of the first number) + 7 (tens digit of the second number) × 3 (units digit of the first number + 2 (carry over from the units’ digit calculation) = 32 + 21 + 2 = 55.

3. Step 3: Finding the hundreds’ place digitMultiply the tens digit of both numbers and add previous carry to it.

Multiply a three digit number to a two digit number

• Step 1:Get the units digit by multiplying the units terms of both numbers. here, we get 8 * 7 = 56 so keep 6 and carry forward 5 to the next operation.

• Step 2:The next digit is obtained by cross multiplying the digits as shown and adding any carry. For this we get 8*7 + 7*2 + 5 = 75. So we carry forward 7 and leave 5 behind.

• Step 3:for the next digit we cross multiply as shown and add the carry if any. here it is 28 + 14 + 7 = 49. So 9 is kept and 4 is the carry.

• Step 4:Finally we get the final digit. Here we get 4 * 7 + 4 = 32.

NumberSquareCube
111
248
3927
41664
525125
636216
749343
864512
981729
101001000
111211331
121441728
131692197
141962744
152253375
162564096
172894913
183245832
193616859
204008000
21441
22484
23529
24576
25625
26676
27729
28784
29841
30900

### Square Root

What is the number you want to find the square root of? Here’s one we’ll use:

46656
First, divide the number to be square-rooted into pairs of digits, starting at the decimal point. That is, no digit pair should straddle a decimal point. (For example, split 1225 into “12 25” rather than “1 22 5”; 6.5536 into “6. 55 36″ rather than”6.5 53 6”.)

Then you can put some lines over each digit pair, and a bar to the left, somewhat as in long division.

+— —- —-
| 4 66 56
Find the largest number whose square is less than or equal to the leading digit pair. In this case, the leading digit pair is 4; the largest number whose square is less than or equal to 4 is 2.

Put that number on the left side, and above the first digit pair.

2
+— —- —-
2 | 4 66 56
Now square that number, and subtract from the leading digit pair.

2
+— —- —-
2 | 4 66 56
|-4
+—-
0
Extend the left bracket; multiply the last (and only) digit of the left-hand number by 2, put it to the left of the difference you just calculated, and leave an empty decimal place next to it.

2
+— —- —-
2 | 4 66 56
|-4
+—-
4_ | 0
Then bring down the next digit pair and put it to the right of the difference.

2
+— —- —-
2 | 4 66 56
|-4
+—-
4_ | 0 66
Find the largest number to put in this blank decimal place such that that number, times the number already there plus the decimal place, will be less than the current difference. For example, see if 1 * 41 is ≤ 66, then 2*42 ≤ 66, etc. In this case it’s a 1. Put this number in the blank you left, and in the next decimal place on the result row on the top.

2 1
+— —- —-
2 | 4 66 56
|-4
+—-
41 | 0 66
Now subtract the product you just found.

2 1
+— —- —-
2 | 4 66 56
|-4
+—-
41 | 0 66
|- 41
+——–
25
Now, repeat as before: Take the number in the left column (here, 41) and double its last digit (giving you 42). Copy this below in the left column, and leave a blank space next to it. (Double the last digit with carry: for example, if you had not 41 but 49, which is 40+9, you should copy down 40+18 which is 58.) Also, bring down the next digit pair on the right.

2 1
+— —- —-
2 | 4 66 56
|-4
+—-
41 | 0 66
|- 41
+——–
42_ 25 56
Now, find the largest digit (call it #) such that 42# * # ≤ 2556. Here, it turns out that 426 * 6 = 2556 exactly.

2 1 6
+— —- —-
2 | 4 66 56
|-4
+—-
41 | 0 66
|- 41
+——–
426 | 25 56
|- 25 56
+————-
0
When the difference is zero, you have an exact square root and you’re done. Otherwise, you can keep finding more decimal places for as long as you want.

Here is another example, with less annotation.

7 . 2 8 0 1 …
+———————-
7 | 53 . 00 00 00 00 00
| 49
+———————-
142 | 4 00
| 2 84
+———————-
1448 | 1 16 00
| 1 15 84
+———————-
14560 | 16 00
| 0
+———————-
145601 | 16 00 00
| 14 56 01
+———————-
| 1 43 99 00

### Solved Question Papers

Q.
248+51+169−−−√−−−−−−−−−√−−−−−−−−−−−−−−−√

16
17
26
18

Ans .16

Explanation :
248+51+13−−−−−−√−−−−−−−−−−−−−√=248+64−−√−−−−−−−−−√=256−−−√=16

Q.
If a * b * c = (a+2)(b+3)√c+1 find the value of 6 * 15 * 3\)

2
3
4
6

Ans .3

Explanation :
(6+2)(15+3)√3+1=8∗18−−−−−√/4=144−−−√/4=3

Q.
Find the value of 25/16−−−−−√

3/4
9/4
5/4
7/4

Ans .5/4

Explanation :
25/16−−−−−√=5/4

Q.
What is the square root of 0.0009?

0.003
0.0003
0.09
0.03

Ans .0.03

Explanation :
0.0009−−−−−√=91000−−−−√=3/100=0.03

Q.
What will come in place of question mark ? 32.4/?−−−−−√=2

8.21
8.31
8.41
8.1

Ans .8.1

Explanation :
32.4/x−−−−−−√=2;Then,32.4/x=4<=>4x=32.4<=>x=8.1

Q.
What will come in place of x in 86.49−−−−√+5+x2−−−−−√=12.3

2.2
3.2
2.23
2.0

Ans .2

Explanation :
86.49−−−−√+5+x2−−−−−√=12.3;9.3+5+x2−−−−−√=12.3;5+x2−−−−−√=12.3−9.3=3;5+x2=9;x2=9−5;x=2

Q.
Find the value of 0.289/0.00121−−−−−−−−−−−√
171 / 11
173 / 11
175 / 11
170 / 11

Ans .170 / 11

Explanation :
√0.289/0.00121 = √0.28900/0.00121 = √28900/121 = 170 / 11

Q.
If 1+(x/144)−−−−−−−−−−√=13/12, the find the value of x

25
35
45
55

Ans .25

Explanation :
√1 + (x / 144) = 13 / 12 -> ( 1 + (x / 144)) = (13 / 12 )2
= 169 / 144 -> x / 144 = (169 / 144) – 1 ; -> x / 144 = 25/144 -> x = 25.

### Quiz

Score more than 80% marks and move ahead else stay back and read again!

Q1: Squares of 16,17,18,19
256,289,324,361
256,288,324,361
225,289,324,361
256,289,344,361

ANS.1

Q2: Which number should be multiplied with 3267 to make it a perfect cube
11
9
7
13

ANS.1

Q3: Cube of 9
729
749
769
829

ANS.1

Q4: Cube of 7
243
363
393
343

ANS.4

Q5: An employer pays Rs.20 for each day a worker works and for forfeits Rs.3 for each day is idle at the end of sixty days a worker gets Rs.280 . for how many days did the worker remain ideal?
20
40
35
30

ANS.2