## Square root and Cube root

**Important points to remember:**

**Square root:**If a

^{2}= 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

^{3}a

**3)**ab = a × b

4) (a/b) = | a | = | a | × | b | = | ab |

b | b | b | b |

**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?**

**) Finding the square of large numbers**

Example: 47

^{2}= 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}= a

^{2}+ 2ab + b

^{2}

Here, (4 + 7)

^{2}= 4

^{2}+ 2 × 4 × 7 + 7

^{2}

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.

94

^{2}< 8888 < 95

^{2}

**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

94

^{2}< 8888 < 95

^{2}

**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

Correct Option: (c)

**Remember the trick discussed in Quick Tips and Tricks**

**Step 1:**Split the number 59 29

7

^{2}= 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

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.

**Q 3.**Find the value of 151

a. 12.459

b. 12.292

c. 13.591

d. None of these

Correct Option: (b)

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

**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. 12151 | = | 7 | = 12.583 |

12 | 12 |

**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**

a. 9

b. 8

c. 6

d. 4

View solution

**Q 4.**Find is the value ofa. 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

Square root of 256 is 16

Square root of 81 is 9

Square root of 36 is 6

10 + 6 = 16 = 4

a. 0.3

b. 0.7

c. 0.09

d. None of these

**Q 5.**Find the value ofa. 0.3

b. 0.7

c. 0.09

d. None of these

View solution
Correct Option: (a)
Step 1: First find 0.000729Step 2:The value of | ||||||||||||||||||||||||||||||||||||

Type 2: Find the missing number
Q 6. 19.36 + 9 + (?)^{2} = 9.4a. 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 19.3619.36 = 4.4 Step 2: 4.4 + 9 + (x) ^{2} = 9.49 + (x) ^{2} = 5Squaring both the sides, we get 9 + x ^{2} = 25x ^{2} = 16x = 4
a. 9 × 10 ^{–3}b. 9 × 10 ^{–5}c. 81 × 10 ^{–4}d. 81 × 10 ^{–5}View solution
Correct Option: (d)
a. 659 b. 694 c. 841 d. 859 View solution
Correct Option: (c)
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,
Q 10. If 3^{n} = 2187, then , find the value of na. 7 b. 9 c. 11 d. 14 View solution
Correct Option: (d)
2187 = 3 ^{7}3 ^{n} = 3^{7}Squaring both the sides, we get 3 ^{n} = (3^{7})^{2}3 ^{n} = 3^{14}Therefore, n = 14
a. 4.236 b. 2.236 c. 3.346 d. 1.566 View solution | ||||||||||||||||||||||||||||||||||||

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
2 ^{2} × 2^{2} × 3 × 11^{2}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.
Step 1: Find the factors of 1470Alternate Solution:
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.
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 as | 326 |

10 ^{6} |

18

324 < 326 < 361

We have to find least number to be subtracted, hence

Given number – smaller number

326 – 324 = 2

^{2}< 326 < 19^{2}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} |

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