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Square Roots and Cube Roots

Introduction

Square Root and Cube Root Table

Square Root

Cube Root

Solved Problems

  1. Numbers

  2. HCF and LCM of Numbers

  3. Fractions

  4. Simplification

  5. Square Roots and Cube Roots

  6. Average

Square Root and Cube Root are the important concepts used in Mathematics. As we have seen square numbers can be represented by a two-dimensional square, a cube number can be represented by a three-dimensional cube.

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We calculate the area of a square as side x side i.e., side , but if we have to calculate the side of a square, we need to take the square root of the area. Thus, we can say that the square root is the inverse operation of squaring a number.

 

Similarly, when we find the volume of the cube, we have volume = side , but if we want to find the side of a cube, we have to take the cube root of the volume. Thus, we can say that the cube root is the inverse operation of cubing a number.

The square root and cube root symbols are √ and ∛ respectively.

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Examples:

  • √4 = √(2 × 2) = 2

  •  âˆ›27 = âˆ›(3 × 3 × 3) = 3

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Square Root and Cube Root Table

                           Memorizing the squares and the square roots of the first few numbers are almost elementary and it can help you to solve problems much faster rather than having to work on it. Following is the square roots list and Cube root list of the first 15 natural numbers.

Square Root

               The square root of a number x is that number which when multiplied by itself gives the number x itself. The number x is a perfect square.

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For Example, 2  =4, or the square root of 4 is 2

                   3  =9, or the square root of 9 is 3

                   4  = 16, or the square root of 16 is 4

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The symbol of the square root is √

                                Therefore, the square root of 4 is represented as √4 = 2.

                                        And the square root of 9 is represented as √9 = 3 and so on.

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Properties of Square Root:

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  • If the unit digit of a number is 2, 3, 7, and 8 then its square root is not a natural number.

  • If a number ends in an odd number of zeros, then its square root is not a natural number.

  • The square root of an even number is even and that of an odd number is odd.

  • Negative numbers have no squares root in a set of real numbers.

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Mathematics Tricks for Square Roots

There are different methods and tricks to find the square root of a number. They are:

  • Using Long Division Method

  • The square root of perfect squares using the prime factorization method.

  • Approximation of square root of a number for not a perfect square number.

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Here, the easiest method trick to find the square root of a number is given below:

                               In order to calculate the square root, we first need to find the factors of a given number, then group the common factor together. Group the pairs separately if the factors have any perfect square. The square root of the square of a number is the number itself.

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For example, the number 36 The factors of 36 is given as 6 x 6. Since it is a perfect square, its square root is 6.

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Cube Root

                  The cube root of a number a is that number which when multiplied by itself three times gives the number ‘a’ itself.  

 

For Example, 2  =8, or the cube root of 8 is 2

                   3  = 27, or the cube root of 27 is 3

                   4  = 64, or the cube root of 64 is 4

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The symbol of the cube root is a    or ∛

                                          Thus, the cube root of 8 is represented as ∛8 = 2

                                                       and that of 27 is represented as ∛27 = 3 and so on.

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Properties of Cube Root

  • Cube root of all the odd numbers is an odd number. For example: ∛27 = 3, ∛125 = 5.

  • Cube root of all the even natural numbers is even. For example: ∛8 = 2, ∛64 = 4.

  • The cube root of a negative integer always results in negative.

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Ex. 1. Evaluate √6084 by factorization method .
   

Sol. Method: Express the given number as the product of prime factors.                               2 6084
                   Now, take the product of these prime factors choosing one out of                     2 3042
                   every pair of the same primes. This product gives the square root                     3 1521
                   of the given number.                                                                                   3 507
           Thus, resolving 6084 into prime factors, we get:                                                  13 169
                                                                          6084 = 22 x 32 x 132                           13
                                                                        √6084 = (2 x 3 x 13)

                                                                                   = 78.

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2. Evaluate: √248 + √51 + √ 169 .


Sol. Given expression

                                 = √248 + √51 + 13

                                 = √248 + √64

                                 = √ 248 + 8

                                 = √256

                                 = 16.

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