Square roots of numbers

Square roots of numbers

          It is customary not to write ‘+’ sign before a positive number. For example, +10 = 10. 

We shall, henceforth, observe this convention in all discussions.

6 ×  = 36     ∴ 6 is the square root of 36.

Similarly, (-6) × (-6) = 36       ∴ (-6) too is a square root of 36.

          
Thus, 6 and -6 are the two square roots of 36.

         

study the following video lessons:

 In the same way, work out in your mind the pairs of numbers of which 4, 49 and 121 are the squares and write statements like the above.

          What do we learn by observing the numbers and their square roots in the above examples?
we see that –

* Every positive number has two square roots.

* These square roots are opposite numbers of each other.

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Lowest Common Multiple (LCM)

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GO THROUGH EXAMPLE VIDEO AND

 LET US FIND ANSWER THE QUESTION BELOW

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Lowest Common Multiple 

(LCM)

Carefully consider the following numbers that are divisible by 8 and by 12.

Numbers divisible by 8 (i. e. multiples of 8) :

8, 16, 24, 32, 40, 48, 56, 64, 72, …

Numbers divisible by 12 (i. e. multiples of 12) :

12, 24, 36, 48, 60, 72, 84, ….

Here the numbers 24, 48, 72, .. are the common multiples of 8 and 12. 24 is the smallest or the lowest of them all. 

Thus, 24 is the smallest or the lowest common multiple or LCM of  8 and 12.

·       Note that the LCM of relative prime numbers is the product of those numbers. 

Examples:

  Find the LCM :

 65, 39

Greatest Common Divisor (GCD)

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Greatest Common Divisor (GCD)

let us consider the divisors of 84 and 48.

Divisors of 84 : 1 , 2 , 3 , 4 , 6 , 7 , 12 , 14 , 21 , 28 , 42, 84

Divisors of 48 : 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

We find that the underlined factors 1, 2, 3, 4, 6, 12 are present in both groups. They are called common divisors.

Of these common divisors 1, 2, 3, 4, 6, 12 the biggest divisor is 12.

 It is called the greatest common divisor or the GCD for short. It is also called highest common factor or HCF.

view the examples:

Find the GCD :

 120, 96

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Indices

CHAPTER 8.

Indices

               Consider 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2. Here , the multiplication of the number 2 by itself is shown by writing it nine times. 

This multiplication is written as 2⁹. The number ‘2⁹’ is as exponential number .

           2⁹  is read as ‘ 2 raised to power 9’ or ‘2 to the power 9’ or ‘2 raised to 9’.

            

In the number 2⁹ , 2 is called the based of the number and 9 is called the index or exponent.

	Multiplication form	Index form	Base	Index
1.	7 × 7 × 7	73	7	3
2.	3 × 3 × 3 × 3 × 3	35	3	5
3.	9 × 9	92	9	2
4.	2 × 2 × 2 × 2 × 2 × 2	26	2	6
5.	1 × 1 × 1 × 1 × 1	15	1	5

Example : study the following table :  

2³  = 2 × 2 × 2

2² = 2 × 2

2¹ = 2

2³ is read as the third power of two or the cube of two.

2² is read as the second power of two or the square of two.

2 ¹ = 2, i.e. the first power of a number is equal to the number itself. Therefore, if the index of a number is 1, it is the custom not write it.

       

 For example, 5 is the same as 5¹, 10 means 10¹, etc.