Square Root and Cube Root - Aptitude test, questions, shortcuts, solved example videos

Square Root and Cube root - Aptitude test, questions, shortcuts, solved example videos

Square root and Cube root

Important points to remember:

Square root: If a2 = b, we say that the square root of b is a
It is written as b = a

2) Cube root: Cube root of a is denoted as 3a

3) ab = a × b
4) (a/b) = a = a × b = ab
bbbb

5) Number ending in 8 can never be a perfect square.

6) Remember the squares and cubes of 2 to 10. This will help in easily solving the problems.



Find the numbers which have same digit in unit’s place:



Quick Tips and Tricks

1) Finding square root of 5, 4 and 3 digit numbers

How to find the square root of 5 digit number ?





How to find the square root of 4 digit number?





How to find the square root of 3 digit number?





2) Finding the square of large numbers

Example: 472 = 2209

Square of 47 can be easily determined by following the steps shown below:

Step 1: Split the number 47 as 4 and 7.
Step 2: Use the formula: (a + b)2 = a2 + 2ab + b2
Here, (4 + 7)2 = 42 + 2 × 4 × 7 + 72
Without considering the plus sign, write the numbers as shown below:
[16] [56] [49]



Step 1: Write down 9 from 49 and carry 4 to 56. [-----9]
Step 2: After adding 4 to 6, we get 10. Therefore, write down zero and carry 1 (5 + 1 = 6) to 16. [----09]
Step 3: 6 + 6 = 12, write down 2 and carry one. [---209]
Step 4: Finally write the answer along with (1 + 1 = 2). [2209]

3) Finding the cube root of 6 digit number?

Note: Cube roots of 6, 5, 4 or 3 digit numbers can be easily found out by using the same trick as used to find the square root of larger digits.

Example: 3132651

Remember: The last 3 numbers are to cut off and the nearby cube of first remaining numbers is to be found out.

Step 1: Split the number 132 and 651
Step 2: 125 is the cube of 5, which is the closest number to 132. Hence, first number i.e. the number in ten’s place is 5.
Step 3: 1 is the digit in unit’s place. Hence, the digit in unit’s place is 1.
Hence, the cube root of 132651 is 51.

4) How to find a number to be added or subtracted to make a number a perfect square ?

For easy understanding, let’s take an example.

Example: 8888



Step 1: Divide 8888 by 9. We get remainder 7.
Step 2: Add Divisor and Quotient [9 + 9 = 18]
Step 3: Now the next divisor will be (18 and number x) which will divide the next dividend. In this case, 4 is the number x and now the divisor becomes 184 × 4 =736.
Step 4: This step is to be followed depend the number of digits in the dividend.

Case 1: If we have to find a number to be added to make a number perfect square, then

Consider a number greater than the quotient. Her quotient is 94, hence consider 95.
942 < 8888 < 952
8836 < 8888 < 9025
Number to be added = Greater number – Given number
Number to be added = 9025 – 8888 = 137

Case 2: If we have to find a number to be subtracted to make a number perfect square, then

942 < 8888 < 952
8836 < 8888 < 9025
Number to be subtracted = Given number - Smaller number
Number to be added = 8888 – 8836 = 52

Question Variety

There are basically 4 types of questions asked from this chapter. Understanding and practicing each of the 4 types will help you deal successfully with the problems from this chapter.

Type 1: Find the square root and cube roots of given numbers


Q 1. Find the square root of 5929

a. 49
b. 33
c. 77
d. 73
View solution

Correct Option: (c)

Remember the trick discussed in Quick Tips and Tricks

Step 1: Split the number 59 29
72 = 49 is the nearest number to 59.Hence, the digit in ten’s place is 7.

Step 2: Last digit of number 29 is 9. Therefore, 3 or 7 are the digits in unit’s place.
Multiply 3 by next consecutive higher number i.e. 4
3 × 4 = 12
But 12 < 59, hence consider the largest number among 3 and 7.
The digit in unit’s place is 7.
Hence, the square root of 5929 is 77


Q 2. Find the cube root of 1728.

a. 12
b. 14
c. 16
d. 18
View solution

Correct Option: (a)

Hint: This type of questions can be easily solved by resolving the given number as the product of prime factors and select one common factor among the repeated factors.

The number, 1728 is easily divisible by 2, 4, 8, 12, etc. Selecting the greater number 12 will reduce the taken consumed to solve the problem.


1728 = 12 × 12 × 12 = 123


Q 3. Find the value of 151

a. 12.459
b. 12.292
c. 13.591
d. None of these
View solution

Correct Option: (b)

Hint:
Trick to find the square root of numbers which are not perfect squares.

Step 1: Find the closet square to 151. 144 is the closest square, its square root is 12.

Step 2: Now divide the given number by the square root of closest square i.e. 12

151 = 7 = 12.583
1212

Step 3: Take the average of 12.583 and the square root of the closest number.
12 + 12.583 = 12.2915 = 12.292
2
151 = 12.292 is the approximate value upto 3 decimal places


Q 4. Find is the value of


a. 9
b. 8
c. 6
d. 4
View solution

Correct Option: (d)



Square root of 256 is 16



Square root of 81 is 9



Square root of 36 is 6
10 + 6 = 16 = 4


Q 5. Find the value of


a. 0.3
b. 0.7
c. 0.09
d. None of these
View solution

Correct Option: (a)


Step 1: First find 0.000729


Step 2:
The value of

Type 2: Find the missing number


Q 6. 19.36 + 9 + (?)2 = 9.4

a. 3.5
b. 4
c. 4.4
d. 5
View solution

Correct Option: (b)

Let the unknown number be x.

Step 1: Firstly, find the value of 19.36
19.36 = 4.4

Step 2:
4.4 + 9 + (x)2 = 9.4
9 + (x)2 = 5

Squaring both the sides, we get
9 + x2 = 25
x2 = 16
x = 4

Q 7. If 0.09 × 0.9 × a = 0.009 × 0.9 × b, thena is _______
b

a. 9 × 10–3
b. 9 × 10–5
c. 81 × 10–4
d. 81 × 10–5
View solution

Correct Option: (d)

The given question can be written in the form of a
b
a = 0.009 × 0.9
b0.09 × 0.9
Squaring both sides, we get
a = 0.009 × 0.9 × 0.009 × 0.9 = 0.0000729 = 0.007290.00081 = 81 × 10 –5
b0.09 × 0.90.099

Q 8. 28x + 1426 = 3 of 2984. Find x
4

a. 659
b. 694
c. 841
d. 859
View solution

Correct Option: (c)

28x + 1426 = 3 of 2984.
4
28x + 1426 = 3 × 2984= 2238
4
28x = 2250 – 1426 = 812
x = 29
x = 841

Type 3: Find the value of _________


Q 9. If 15625 = 125, then find the value of (156.25 + 1.5625 + 0.015625 + 0.00015625)

a. 1.38875
b. 13.8875
c. 138.875
d. 1388.75
View solution

Correct Option: (b)

Write the given expression without decimal points.


We are given, 15625 = 125

Therefore,

125 + 125 + 125 + 125 = 12.5 + 1.25 + 0.125 + 0.0125 = 13.8875
10100100010000


Q 10. If 3n = 2187, then , find the value of n

a. 7
b. 9
c. 11
d. 14
View solution

Correct Option: (d)

2187 = 37
3n = 37

Squaring both the sides, we get
3n = (37)2
3n = 314
Therefore, n = 14

Q 11. If 5 = 2.236, then find the value of 3201125180
5

a. 4.236
b. 2.236
c. 3.346
d. 1.566
View solution

Correct Option: (b)

= 320 – (1/5) 125180
= 64 × 5 – (1/5) 25 × 536 × 5
= 85 – (1/5)55 – 65
= 85 – (1 + 6) 5
= 5
Value of 5 = 2.236

Type 4: Find the least number to make a number perfect square.


Q 12. Find the least number by which 5808 must be multiplied to make it a perfect square.

a. 3
b. 4
c. 6
d. 7
View solution

Correct Option: (a)

Step 1: Find the factors of 5808


Factors of 5808 are 2 x 2 x 2 x 2 x 3 x 11 x 11
22 × 22 × 3 × 112
Step 2: Therefore, to make 5808 a perfect square it must be multiplied by 3, because 3 is non-repetitive here. Square root of number is determined by considering the common factor only once as shown below.
5808 x 3 = 17424
Hence, after multiplying 5808 by 3 we get 17424 which is a perfect square of 132
2 x 2 x 3 x 11 = 132
17424 = 132


Q 13. Find the least number by which 1470 must be divided to get a number which is a perfect square.

a. 6
b. 9
c. 12
d. 30
View solution

Correct Option: (d)

This numerical can be easily solved, if we solve it by considering the options.
Divide 1470 by each option and find out the number which is perfect square.

1470 = 245, 1470 = 163.331470122.51470 = 49
691230
Hence, it is clear that the correct option is d = 30

Alternate Solution:

Step 1: Find the factors of 1470



1470 = 7 × 7 × 5 × 6
We are asked to find the number by which 1470 must be divided to get a number which is a perfect square.
Step 2: Hence, we have to divide the number by 5 x 6, as these factors are non-repetitive.
1470 = 49
5 × 6
49 is a perfect square of 7


Q 14. Find the least number to be subtracted from 0.000326 in order to make it a perfect square.

a. 0.0004
b. 0.04
c. 0.000002
d. 0.0002
View solution

Correct Option: (c)

0.000326 can be written as326
10 6
Now find the least number to be subtracted from 326 to make it a perfect square.


182 < 326 < 192
324 < 326 < 361
We have to find least number to be subtracted, hence
Given number – smaller number
326 – 324 = 2
Hence, the number is 2 = 0.000002
10 6


Q 15. Find the smallest number by which 1780 must be added to make the sum a perfect square.

a. 46
b. 49
c. 69
d. 72
View solution

Correct Option: (c)

Step 1: Find the square root of 1780
1780 = 42.190
Take the square of next number after 42 i.e. 43.
Step 2: Find the least number
Subtract given number from 432
432 – 1780 = Number to be added
1849 – 1780 = 69
Hence, the number to be added is 69.

Now, cross-check your answer by adding 69 and 1780
1780 + 69 = 1849
1849 = 43
It is a perfect square.

Alternate solution


422 < 1780 < 1849
1764 < 1780 < 1849
Hence, we have to subtract the given number from 1849 to find the number to be added.
1849 – 1780 = 69





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