Powers & Roots

                                                      POWERS AND ROOTS                                              

ð  If ⁿ   multiplied n times, where   is the base, n the exponent and ⁿ an exponential expression.

                                   ⁿ is read as ‘‘  to the power n’’.

ð  Power 2 = SQUARE and power 3 = CUBE, i.e., ²  is read ‘‘ a square ’’ , ³  is read as ‘‘ a cube ’’.

Power of (-1) –

(i) ² = ⁴ =  

              i.e., (-1) raised to an even positive integral power = 1.

(ii) (-1)¹ = -1; (-1)³ =

              i.e., (-1) raised to an odd positive integral power = -1.

Laws of Exponents –

      Law1-   ^{m n} = ^m+n where  is a non-zero number and m, n are integers.

e.g., 3³ 3⁴ = 3³⁺⁴ = 3⁷ ; 7^-2 7⁵ = 7^-2+5 = 7³ .

      Law2-  ^{m n} =  = , where   is a non-zero number where m, n are integers.

e.g., .

      Law3- , where  is any non-zero number.

e.g.,

      Law 4- where  is any non-zero number.

e.g.,  .

Squares - If a number is multiplied by itself, the product so obtained is called the square of thatnumber.

                                e.g.- i.e., 4 is the square of 2.

                                       
 is the square of

ð  Squares of even numbers are always even.

e.g.-

ð  Squares of odd numbers are always odd.

e.g.-  

Perfect squares- 1, 4, 9, 16, 25, 36, etc. , are squares of natural numbers 1, 2, 3, 4, 5, 6, respectively. Such numbers are called perfect squares.

ð  The number of zeros at the end of a square number is double the number of zeros at the end of a given number, i.e., the number of zeros at the end of a perfect square is always even.

Square Roots-   

a.       The square root of a positive integer n is an integer whose square is n.

e.g.- 4 is the square of 2.

ð  2 is the square root of 4.

                                         is the square of

ð   is the square root of .

b.       Since  and also , therefore the number 49 has in fact two square roots 7 and -7, one positive and one negative, same value but opposite sign.

        The symbol  is used to indicate square root. Thus,  and so on. When you are asked to find the square root of a given number, then you shall at this stage find the square root only.

c.       Since  Also since

d.       Table of square roots of perfect squares up to 100.

Number	Square Root	Number	Square Root
1 4 9 16 25 36 49 64 81 100	1 2 3 4 5 6 7 8 9 10	121 144 169 196 225 256 289 324 361 400	11 12 13 14 15 16 17 18 19 20

  

Finding Square Root by Prime Factorisation-

Step1- Split the given number into its prime factors. To do so, start dividing the number by its lowest prime           factor and continue the process.

                Step2- Form pairs of like prime factors.

                Step3- Form each pair, pick out one prime number.

                Step4- Multiply the factors so picked up. The product is the square root of the given number.

                                                e.g.- Find the square root of 576.

                                                                 

                                                               

                                               

2	576
2	288
2	144
2	72
2	36
2	18
3	9
	3

Note:

                                                                                                   

Cubes- The number obtained on multiplying a given number by itself three times is called the cube of the given number.

                                e.g.-

                                        

                                i.e., the cube of a positive number is always positive and the cube of a negative number is always    

                                negative.

Cube Root- Cube root of a number is that number which when multiplied by itself three times gives the original number.

                The symbol  is used to indicate the cube root of a number.

                                e.g.-

ð  Cube root of a negative number is always negative.

Number	1	8	27	64	125	216	343	512	729	1000	1331	1728	2197	2744	3375	4096
Cube Root	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16

Finding Cube Root by Prime Factorisation-

                Step1- Split the given number into its prime factors.

                Step2- Form groups in triplets of the same prime factor.

                Step3- From each triplet, pick out one prime factor.

                Step4- Multiply the factors so picked up. The product is the cube root of the given number.

                                e.g.- 9261= 3 3    7 7 7

                               

                                               

3	9261
3	3087
3	1029
7	343
7	49
	7