3. Indices and Cube root Let’s recall. In earlier standards, we have learnt about Indices and laws of indices. • The product 2 × 2 × 2 × 2 × 2, can be expressed as 25, in which 2 is the base, 5 is the index and 25 is the index form of the number. • Laws of indices: If 푚 and 푛 are integers, then 푚 푛 푚+푛 (i) 푎 × 푎 = 푎 푚 푛 푚 − 푛 (ii) 푎 ÷ 푎 = 푎 iii) (푎 × 푏)m = 푎m × 푏m ° iv) 푎 = 1 1 푎−푚 v) = 푎푚 푚 푛 푚푛 vi) (푎 ) = 푎 푎 푎푚 ( )m vii) 푏 = 푏푚 푎 푏 ( )-m ( )m viii) 푏 = 푎 • Using laws of indices, write proper numbers in the following boxes. 7 -2 20 (i) 35 x 32 = 3 (ii) 37 ÷ 39 = 3 (iii) (34)5 = 3 -3 1 0 1 1 5 (iv) 5 = 5 3 (v) 5 = (vi) 5 = 2 2 5 5 5 7 2 ( )3 ( )-3 3 (vii) (5 x 7) = 5 x 7 (viii) 7 = (ix) 7 = 7 5 Meaning of numbers with rational indices – 1 (i) Meaning of the numbers when the index is a rational number of the form 푛. Let 1 1 1 1 us see the meaning of indices in the form of rational numbers such as 2 , 3 5, ….푛 To show the square of any number, the index is written as 2 and to show the square 1 root of a number, the index is written as 2. Ex. Square root of 25, is written as √25 using the radical sign ‘√ ’. 1 1 Using index, it is expressed as 252. ∴ √25 = 252. 2 In general, square of a can be written as 푎2 and square root of a is written as √푎 or 1 √푎 or 푎2. Similarly cube of 푎 is written as 푎3 1 3 Cube root of 푎 is written as √푎 or 푎3 For example: 3 × 3 × 3 × 3 × 3 = 35 = 243 That is, the 5th power of 3 is 243. 1 5 Conversely 5th root of 243 is expressed as (243)5 or √243 . 1 Hence (243)5 = 3 1 In general 푛th root of 푎 is expressed as 푎푛. Practice Set 3.1 1. Express the following numbers in index form. (1) Fifth root of 13 (2) Sixth root of 9 (3) Square root of 256 1 1 1 Ans. 135 Ans. 96 Ans. 2562 (4) Cube root of 17 (5) Eighth root of 100 (6) Seventh root of 30. 1 1 1 Ans. 173 Ans. 1008 Ans. 307 2. Write in the form ‘풏th root of a’ in each of the following numbers. 1 1 (1) (81)4 (2) 492 Ans. Fourth root of 81. Ans. Square root of 49. 1 1 (3) (15)5 (4) (512)9 Ans. Fifth root of 15 Ans. Ninth root of 512 1 1 (5) 10019 (6) (6)7 Ans. Nineteenth root of 100 Ans. Seventh root of 6. 풎 . (ll). The meaning of numbers, having index in the rational form 풏 Ex. 82 = 64 1 1 Cube root of 64 = (64)3 = (82)3 = 4 Cube root of 8 = 4 ...... (I) 1 similarly, cube root of 8 cube 83 = 2 1 ∴ Square of cube root of 8 is (83)2 = 22 = 4 …….. (II) From (I) and (II) 1 1 2 Cube root of square 8 = Square of cube root of 8. Using indices, (8 )3= (83)2. Rules for rational indices are the same as those for integral indices. 1 1 2 2 Using the rule (푎m) = 푎mn we get (8 )3= (83)2 = 83 2 From this we get two meanings of the number 83. 2 1 (i) 83 = (82)3 i.e. cube root of square of 8. 2 1 (ii) 83 = (83)2 i.e. Square of cube root of 8. 4 1 4 Similarly, 275 = (27 )5 that means fifth root of fourth power of 27’, 4 1 4 And 275 = (275) means ‘fourth power of fifth root of 27’. 푚 Generally we can express two meanings of the number 푎 푛 푚 1 푛 m 푛 th th 푎 = (푎 ) means 푛 root of 푚 power of a’. 푚 1 푎 푛 = (푎푛)m means 푚th power of 푛th root of 푎. Practice Set 3.2 1. Complete the following table. Sr.No. Number Power of the root Root of the power 3 (1) (225)2 Cube of square root of 225 Square root of cube of 225 4 th th (2) (45)5 Fourth power of fifth root of 45 5 root of 4 power of 45 6 th th th (3) (81)7 Sixth power of 7 root of 81 7 root of 6 power of 81 4 th th th (4) (100)10 Fourth power of 10 root of 100 10 root of 4 power of 100 3 th th rd (5) (21)7 Third power of 7 root of 21 7 root of 3 power of 21 2. Write the following numbers in the form of rational indices. (1) Square root of 5th power of 121. 5 Ans. (121)2 (2) (2) Cube of 4th root of 324. 3 Ans. (324)4 (3) 5th root of square of 264. 2 Ans. (264)5 (4) Cube of cube root of 3. 3 Ans. (3)3. Cube and Cube Root – If a number is written 3 times and multiplied, then the product is called the cube of the number. Ex. 6 × 6 × 6 = 63 = 216 Hence 216 is the cube of 6. Ex. 1) Find the cube of 17. 173 = 17 × 17 × 17 = 4913 Cube of a positive number will always be a positive. Ex. 2) Find the cube of (-6). (-6)3 = (-6) × (-6) × (-6) = - 216 If the number is negative, then its cube will always be is the negative. 2 (− ) Ex. 3) Find the cube of 5 2 2 2 2 (− )3 (− ) (− ) (− ) 5 = 5 × 5 × 5 8 − = 125 Ex. 4) Find the cube of (1.2) (1.2)3 = 1.2 x 1.2 x 1.2 = 1.728 Ex. 5) Find the cube of (0.02). (0.02)3 = 0.02 x 0.02 x 0.02 = 0.000008 To find the cube root We know, how to find the square root of a number by factorisation method. Using the same method, we can find the cube root. Ex. (1) Find the cube root of 216. Solution: First find the prime factor of 216. 216 = 2 x 2 x 2 x 3 x 3 x 3 Each of the factors 3 and 2, appears thrice. So let us group them as given below, 216 = (3 × 2) × (3 × 2) × (3 × 2) = (3 × 2)3 = 63 1 3 ∴ √216 = 6 that is (216)3 = 6 Ex. (2) Find the cube root of -1331. Solution: To find the cube root of -1331, let us factorise 1331 first. 3 1331 = 11 × 11 × 11 = 11 - 1331 = (-11) × (-11) × (-11) 3 = (-11) 3 ∴ √−1331 = - 11 Ex. (3) Find the cube root of 1728. Solution: 1728 = 8 × 216 = 2 × 2 × 2 × 6 × 6 × 6 3 3 3 m m m ∴ 1728 = 2 × 6 = (2 × 6) ….. 푎 × 푏 = (푎 × 푏) 3 √1728 = 2 × 6 = 12. (Note that, cube root of - 1728 is -12.) 3 Ex. (4) Find √0.125. 3 125 3√0.125 = √ Solution: 1000 3 푚 √125 푎 m 푎 = 3 …… ( ) = √1000 푏 푏푚 3 3 1 √5 푚 = 3 …… (푎 )푚 = 푎 √103 1 5 m = = 0.5 … (푎 )푚 = 푎 10 Practice Set 3.3 1. Find the cube roots of the following numbers. Solution: (1) 8000 = 8 × 1000 = 2 × 2 × 2 × 10 × 10 × 10 3 3 = 2 × 10 3 = (2 × 10) 3 = 20 3 Ans. √8000 = 20. (2) 729 = 9 x 81 = 9 x 9 x 9 3 = 9 3 Ans. √729 = 9. (3) 343 = 7 x 49 = 7 x 7 x 7 3 = 7 3 Ans. √343 = 7. (4) -512 Let us first factorise 512 512 = 2 x 256 = 2 x 2 x 128 = 2 x 2 x 2 x 64 = 2 x 2 x 2 x 4 x 16 = 2 x 2 x 2 x 4 x 4 x 4 = 23 x 43 = (2 x 4)3 = 83 3 ∴ -512 = (-8) 3 Ans. √−512 = -8. (5) -2744 2744 is divisible by 2 ∴ it must be divisible by 8 2744 = 8 x 343 = 8 x 7 x 49 = 2 x 2 x 2 x 7 x 7 x 7 = 23 x 73 = (2 x 7)3 = 143 3 Ans. √−2744 = -14. (6) 32768 = 8 x 4096 = 8 x 8 x 512 = 8 x 8 x 8 x 64 = 8 x 8 x 8 x 4 x 4 x 4 = 83 x 43 = (8 x 4)3 = 323 3 Ans. √32768 = 32. 2. Simplify: ퟑ √ ퟐퟕ (1) ퟏퟐퟓ Solution: 27 3 × 3 × 3 33 125 = 5 × 5 × 5 = 53 3 3 27 √33 3 ∴ √ 3 125 = √53 = 5 3 27 ퟑ √ Ans. 125 = ퟓ. ퟑ √ퟏퟔ (2) ퟓퟒ 16 8 × 2 8 23 54 = 27 × 2 = 27 = 33 3 3 16 √23 2 ∴ √ 3 54 = √33 = 3 3 16 2 √ Ans. 54 = 3. ퟑ ퟑ (3) If √ퟕퟐퟗ = 9, then √ퟎ. ퟎퟎퟎퟕퟐퟗ =? Solution: 3 √0.000729 3 729 √ = 1000000 3 √729 9 3 = √1000000 = 100 = 0.09. [Note: There are six decimal places in 0.000729. ∴ in cube root there will be 2 decimal places.] .