**SSC CHSL Topic Wise Study Material – Quantitative Aptitude – Square Root and Cube Root**

Contents

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**Square**

If a number is multiplied with itself, then the result of this multiplication is called the square of that number.

**e.g.,** Square, of 5 = 5² = 5×5 = 25

**Perfect Square**

A natural number n is called a perfect square, if there exists a natural number m, such that n = m².

**e.g.,** The numbers 1,4, 9, 16,… are square numbers.

**Square Root**

The square root of given number is that number whose square is equal to the given number. In general, we can say that, if a² = b, then square root of b is a. To represent the square root of a number, use the symbol ‘√’ on given number.

**Methods to Find Square Root**

The following methods are used to find the square root of any number

**By Prime Factorization Method**

The various steps involved in prime factorization method are

**Step I** Resolve the given number into prime factors.

**Step II** Make pairs of prime factors such that both the factors in each pair are equal.

**Step III** Take one factor from each pair.

**Step IV** Find the product of factors obtained in Step III, which is the required square root.

**By Division Method**

The steps involved in division method are

**Step I** In the given number, place bars over every pair of digits starting with the unit’s digit. Each pair and the remaining one digit (if any) on the extreme left is called a period.

**Step II** Think of the largest number whose square is less than or equal to the first period.

**Step III** Put the quotient above the period and write the product of divisor and quotient just below the first period. .

**Step IV** Subtract the product of divisor and quotient from the first period and bring down the next period to the right of the remainder. This becomes the new dividend.

**Step V** Now, the new divisor is obtained by adding the unit digit of the division to the divisor and annexing with a suitable digit which is also taken as the next digit of the quotient, chosen in such a way that the product of the new divisor and this digit is equal to or just less than the new dividend.

**Step VI** Repeat Steps II, III, IV and V till all the periods have been taken up.Now, the quotient, so obtained is the required square root of the given number.

**Square Root of a Decimal Number**

If the given number is in decimal form, then we separate the digits into periods of two to the right and left starting from the decimal point and then proceed normally. If the number of digit after the decimal are not even, then zero is put after the digits (extreme right) to make the digits even.

**Square Root of a Fraction**

If the given number is in fractional form, then find the square root of numerator and denominator separately and then write them in fractional form. Thus, the resultant fraction is the square root of the fraction.

**Cube**

If a number is multiplied two times with itself, then the result of this multiplication is called the cube of that number.

e.g., Cube of 4 = 4 x 4 x 4=4³

**Cube Root**

The cube root of a given number is the number whose cube is the given number.

In general, we Can say that a³ = b, then cube root of ‘b’ is ‘a’. Cube root of a number ‘a’ is denoted as 3√a.

Following method is used to find the cube root of given number

**Prime Factorization Method**

The various steps involved in prime factorization method are

**Step I** Resolve the given number into prime factors.

**Step II** Group the factors having three numbers in such a way that each number of the group is same.

**Step III** Take one factor from each group.

**Step IV** Find the product of the factors obtained in Step III which is the required cube root.

**Example Find the cube root of 9261.**

(a) 20

(b) 30

(c) 15

(d) 21

**Essential Points**

**Reference Corner**

**1. 9x² + 25 – 30x can be expressed as the square of SSC (10+2) 2015**

(a) 3x²-25

(b)3x+5

(c)-3x-5

(d)3x-5

**Answer:**

(d)9x² +25-30x

= (3x)² – 2 x(3) x (5) x + (5)² = (3x – 5)²

The required value is 3x – 5.

**Practice Exercise**

**1.**

(a) 3

(b) 4

(c) 1

(d) 2

**2.**

(a)5/9

(b)1-1/7

(c)4/7

(d)1-2/7

**3.By what least number should 4320 be multiplied so as to obtain a number which is a perfect cube?**

(a) 40

(b) 50

(c) 60

(d) 80

**4.(3√1000 + 3√0.008 – 3√0.125) is equal to**

(a) 9.7

(b) 9.97

(c) 9.997

(d) 9.9997

**5.If √1+x/961 = 32/31, then the value of x is**

(a) 63

(b) 61

(c) 65

(d) 64

**6.The square root of 0.324 x 0.081 x 4.624/1.5625 x 0.0289 x 72.9 x 64 is**

(a) 24

(b) 2.4

(c) 0.024

(d) 1.2

**7.By what least number should 675 be multiplied so as to obtain a perfect cube number?**

(a) 3

(b) 5

(c) 24

(d) 40

**8.How many perfect squares lie between 120 and 300?**

(a) 5

(b) 6

(c) 7

(d) 8

**9.If (10.15)² = 103.0225, then the value of ****√1.030225 + √10302.25 is**

(a) 1025.15

(b) 103.515

(c) 102.515

(d) 102.0515

**10.Five times of a positive integer is equal to 3 less than twice the square of that number. The number is**

(a) 3

(b) 13

(c) 23

(d) 33

**11.The number whose square is equal to the difference of the squares of the numbers 68 and 32, is**

(a) 36

(b) 48

(c) 60

(d) 64

**12. The sum of digits of the smallest number, which when multiplied by 900, gives a perfect cube is**

(a) 2

(b) 3

(c) 6

(d) 9

**13. The sum of the squares of two positive integers is 100 and the difference of their squares is 28. The sum of the numbers is**

(a) 12

(b) 13

(c) 14

(d) 15

**14.The smallest number added to 680621 to make the sum a perfect square is**

(a) 4

(b) 5

(c) 6

(d) 8

**15.**

(a)7 1/12

(b)5 5/12

(c) 1 1/12

(d) 1 7/12

**16.**

(a)1

(b)2.5

(c)5

(d)25

**17.**

(a)5/6

(b)7 1/6

(c)7 1/3

(d)8 1/3

**18. How many positive integers loss than 1000 are multiples of 11 whose square roots are whole numbers.**

(a) 2

(b) 4

(c) 8

(d) 11

**19. Three numbers are in the ratio of 3 : 2 : 5 and the sum of their squares is 1862. The smallest of these numbers is**

(a) 24

(b) 21

(c) 14

(d) 35

**20. Find the largest number of 5-digit which is a perfect square?**

(a) 99999

(b) 99764

(c) 99976

(d) 99856

**Answers**

**Hints & Solutions**

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