EXPONENTS AND POWERS 9 Exponents nd Powers CHAPTER. Introduction Do you know? Mss of erth is 5,970,000,000,000, 000, 000, 000, 000 kg. We hve lredy lernt in erlier clss how to write such lrge nubers ore conveniently using exponents, s, 5.97 0 kg. We red 0 s 0 rised to the power. We know 5 nd...... ( ties) Let us now find wht is is equl to?. Powers with Negtive Exponents You know tht, 0 0 0 00 0 0 00 0 0 0 0 0 0? Continuing the bove pttern we get,0 0 Siilrly 0 0 0 0 0 00 0 Wht is 0 0 equl to? 0 0 00 00 0 000 0 Exponent is negtive integer. As the exponent decreses by, the vlue becoes one-tenth of the previous vlue.
9 MATHEMATICS Now consider the following. 7 9 7 The previous nuber is divided by the bse. 9 So looking t the bove pttern, we sy You cn now find the vlue of in siilr nner. We hve, 0 or 0 0 0 0 or 0 0 0 or etc. In generl, we cn sy tht for ny non-zero integer,, where is positive integer. is the ultiplictive inverse of. TRY THESE Find the ultiplictive inverse of the following. (i) 0 5 (iii) 7 (iv) 5 (v) 0 00 We lernt how to write nubers like 5 in expnded for using exponents s 0 + 0 + 0 + 5 0. Let us see how to express 5.6 in expnded for in siilr wy. We hve 5.6 000 + 00 + 0 + 5 + + 6 0 00 0 + 0 + 0 + 5 + 0 + 6 0 TRY THESE Expnd the following nubers using exponents. (i) 05.6 56.9 0 0, 0 0 00
EXPONENTS AND POWERS 95. Lws of Exponents We hve lernt tht for ny non-zero integer, n + n, where nd n re nturl nubers. Does this lw lso hold if the exponents re negtive? Let us explore. (i) We know tht nd Therefore, + Tke ( ) ( ) ( ) ( ) ( ) ( ) (iii) Now consider 5 5 ( ) ( ) ( ) + 5 5 5 5 5 5 () 5 5 (iv) Now consider ( 5) ( 5) ( ) 7 ( ) + 5 for ny non-zero integer. 5 is the su of two exponents nd ( ) + ( ) 7 In Clss VII, you hve lernt tht for ny n non-zero integer, n, where nd n re nturl nubers nd > n. ( 5) ( 5) ( 5) ( 5) ( 5) ( 5) ( 5) ( 5) ( 5) TRY THESE ( 5) () In generl, we cn sy tht for ny non-zero integer, n + n, where nd n re integers. ( ) + Siplify nd write in exponentil for. (i) ( ) ( ) p p 0 (iii) 5 6 On the se lines you cn verify the following lws of exponents, where nd b re non zero integers nd, n re ny integers. (i) (iv) b n n ( ) n n (iii) b (b) b (v) 0 Let us solve soe exples using the bove Lws of Exponents. These lws you hve studied in Clss VII for positive exponents only.
96 MATHEMATICS Exple : Find the vlue of (i) (i) 8 Exple : Siplify (i) ( ) 5 ( ) 0 5 6 9 (i) ( ) 5 ( ) 0 ( ) (5 0) ( ) 5 5 ( ) 5 6 5 ( 6) ( n n ) Exple : Express s power with the bse. We hve, ( n + n, Therefore, () ( ) ( ) ( ) 6 [( ) n n ] Exple : Siplify nd write the nswer in the exponentil for. (i) ( 5 8 ) 5 5 ( ) (5) ( 5) 5 (iii) () (iv) ( ) 8 (i) ( 5 8 ) 5 5 ( 5 8 ) 5 5 ( ) 5 5 5 5 0 0 ( ) (5) ( 5) [( ) 5 ( 5)] [00] 00 (iii) (iv) () () ( ) 6 8 6 ( ) 5 5 5 ( ) ( ) ( ) 5 5 [( ) ] ) [using the lw b (b), ] Exple 5: Find so tht ( ) + ( ) 5 ( ) 7 ( ) + ( ) 5 ( ) 7 ( ) + + 5 ( ) 7 ( ) + 6 ( ) 7 On both the sides powers hve the se bse different fro nd, so their exponents ust be equl.
EXPONENTS AND POWERS 97 Therefore, + 6 7 or 7 6 Exple 6: Find the vlue of 9 Exple 7: Siplify (i) (i) 7 5 5 8 8 5. Evlute.. 7 5 5 8 8 5 {9 8} 6 6 7 5 7 5 5 8 5 8 ( 7) ( 5) ( 5) ( 7) 5 8 7 5 5 7 8 5 5 8 8 6 5 8 5 5 EXERCISE. (i) ( ) (iii). Siplify nd express the result in power nottion with positive exponent. (i) ( ) 5 ( ) 8 (iii) ( ). Find the vlue of. 5 5 (iv) ( 7 0 ) 5 (v) ( 7) (i) ( + ) ( ) (iii) n only if n 0. This will work for ny except or. For,... or () n for infinitely ny n. For, ( ) 0 ( ) ( ) ( )... or ( ) p for ny even integer p. + + In generl, b b
98 MATHEMATICS (iv) ( + + 5 ) 0 (v) 8 5. Evlute (i) (5 ) 6 5. Find the vlue of for which 5 5 5 5. 6. Evlute (i) 7. Siplify. 5 t (i) ( t 0) 8 5 0 t 7 5 8 8 5 5 5 7 5 0 5 5 6. Use of Exponents to Express Sll Nubers in Stndrd For Observe the following fcts.. The distnce fro the Erth to the Sun is 9,600,000,000.. The speed of light is 00,000,000 /sec.. Thickness of Clss VII Mthetics book is 0.. The verge dieter of Red Blood Cell is 0.000007. 5. The thickness of hun hir is in the rnge of 0.005 c to 0.0 c. 6. The distnce of oon fro the Erth is 8, 67, 000 (pprox). 7. The size of plnt cell is 0.000075. 8. Averge rdius of the Sun is 695000 k. 9. Mss of propellnt in spce shuttle solid rocket booster is 50600 kg. 0. Thickness of piece of pper is 0.006 c.. Dieter of wire on coputer chip is 0.00000.. The height of Mount Everest is 888. Observe tht there re few nubers which we cn red like c, 888, 6,95,000 k. There re soe lrge nubers like 50,000,000,000 nd soe very sll nubers like 0.000007. Identify very lrge nd very sll nubers fro the bove fcts nd write the in the djcent tble: Very lrge nubers We hve lernt how to express very lrge nubers in stndrd for in the previous clss. For exple: 50,000,000,000.5 0 Now, let us try to express 0.000007 in stndrd for. Very sll nubers 50,000,000,000 0.000007 --------------- --------------- --------------- --------------- --------------- --------------- --------------- ---------------
EXPONENTS AND POWERS 99 7 0.000007 000000 7 6 0 7 0 6 0.000007 7 0 6 Siilrly, consider the thickness of piece of pper which is 0.006 c. 6.6 0 0.006.6 0 0 0000 0.6 0 Therefore, we cn sy thickness of pper is.6 0 c. TRY THESE. Write the following nubers in stndrd for. (i) 0.00000056 0.00000 (iii) 600000 (iv) 50000. Write ll the fcts given in the stndrd for... Copring very lrge nd very sll nubers The dieter of the Sun is. 0 9 nd the dieter of the Erth is.756 0 7. Suppose you wnt to copre the dieter of the Erth, with the dieter of the Sun. Dieter of the Sun. 0 9 Dieter of the erth.756 0 7 9 9 7. 0. 0. 00 Therefore 7 which is pproxitely 00.756 0.756.756 So, the dieter of the Sun is bout 00 ties the dieter of the erth. Let us copre the size of Red Blood cell which is 0.000007 to tht of plnt cell which is 0.000075. Size of Red Blood cell 0.000007 7 0 6 Size of plnt cell 0.000075.75 0 5 6 7 0 6 ( 5) 7 0 7 0 0.7 0.7 Therefore, 5 (pprox.).75 0.75.75.75. So red blood cell is hlf of plnt cell in size. Mss of erth is 5.97 0 kg nd ss of oon is 7.5 0 kg. Wht is the totl ss? Totl ss 5.97 0 kg + 7.5 0 kg. 5.97 00 0 + 7.5 0 When we hve to dd nubers in stndrd for, we convert the into 597 0 + 7.5 0 nubers with the se exponents. (597 + 7.5) 0 60.5 0 kg. The distnce between Sun nd Erth is.96 0 nd the distnce between Erth nd Moon is.8 0 8. During solr eclipse oon coes in between Erth nd Sun. At tht tie wht is the distnce between Moon nd Sun.
00 MATHEMATICS Distnce between Sun nd Erth.96 0 Distnce between Erth nd Moon.8 0 8 Distnce between Sun nd Moon.96 0.8 0 8.96 000 0 8.8 0 8 (96.8) 0 8 9.6 0 8 Exple 8: Express the following nubers in stndrd for. (i) 0.00005 050000 (i) 0.00005.5 0 5 050000.05 0 6 Exple 9: Express the following nubers in usul for. (i).5 0 5 7.5 0 (iii) 0 5 (i).5 0 5.5 00000 5000 7.5 7.5 7.5 0 0 0.00075 0000 (iii) 0 5 5 0 0.0000 00000 Agin we need to convert nubers in stndrd for into nubers with the se exponents. EXERCISE.. Express the following nubers in stndrd for. (i) 0.0000000000085 0.000000000009 (iii) 600000000000000 (iv) 0.0000000087 (v) 860000000. Express the following nubers in usul for. (i).0 0 6.5 0 (iii) 0 8 (iv).000 0 9 (v) 5.8 0 (vi).69 0 6. Express the nuber ppering in the following stteents in stndrd for. (i) icron is equl to 000000. Chrge of n electron is 0.000,000,000,000,000,000,6 coulob. (iii) Size of bcteri is 0.0000005 (iv) Size of plnt cell is 0.000075 (v) Thickness of thick pper is 0.07. In stck there re 5 books ech of thickness 0 nd 5 pper sheets ech of thickness 0.06. Wht is the totl thickness of the stck. WHAT HAVE WE DISCUSSED?. Nubers with negtive exponents obey the following lws of exponents. () n +n (b) n n (c) ( ) n n (d) b (b) (e) 0 (f) b b. Very sll nubers cn be expressed in stndrd for using negtive exponents.