3 Indices and Standard Form
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1 MEP Y9 Prctice Book A Indices nd Stndrd Form. Inde Nottion Here we revise the use of inde nottion. You will lredy be fmilir with the nottion for squres nd cubes, nd this is generlised by defining: Emple Clculte the vlue of: n... n of these () 0 () Emple Copy ech of the following sttements nd fill in the missing number or numbers: ()
2 . MEP Y9 Prctice Book A () Emple () Determine. Determine. Determine. Epress your nswer to in inde nottion. () Eercises. Clculte: () 0 0 (e) 9 (f) (g) (h) (i). Copy ech of the following sttements nd fill in the missing numbers: ()
3 MEP Y9 Prctice Book A (e) (f) (g) (h). Copy ech of the following sttements nd fill in the missing numbers: () (e) (f) (g) (h). Is 0 bigger thn 0?. Is bigger thn?. Is bigger thn?. Copy ech of the following sttements nd fill in the missing numbers: () 9 (e) (f). Clculte: () + + (e) 0 (f) 0 + 9
4 . MEP Y9 Prctice Book A 9. Clculte: ( ) ( ) () + ( ) ( + ) 0. Writing your nswers in inde form, clculte: () 0 0 (e) 0 0 (f). () Without using clcultor, write down the vlues of k nd m. k m Complete the following: (KS/99M/Tier -/P). Lws of Indices There re three rules tht should be used when working with indices: When m nd n re positive integers,. m n m + n. m n m n or m n m n ( m n) ( ). m n m n These three results re logicl consequences of the definition of n, but relly need forml proof. You cn 'verify' them with prticulr emples s below, but this is not proof: ( ) ( ) 0 (here m, n nd m + n 0) 0
5 MEP Y9 Prctice Book A or, Also, (gin m, n nd m n ) ( ) (using rule ) (gin m, n nd m n ) The proof of the first rule is given below: Proof m n... m of these n of these m + n m + n ( ) of these The second nd third rules cn be shown to be true for ll positive integers m nd n in similr wy. We cn see n importnt result using rule : n n n n 0 but n n, so This is true for ny non-zero vlue of, so, for emple,, nd 00 0.
6 . MEP Y9 Prctice Book A Emple Fill in the missing numbers in ech of the following epressions: 9 () ( ) 0 0 () ( ) 0 0 Emple Simplify ech of the following epressions so tht it is in the form n, where n is number: () () + ( ) + ( )
7 MEP Y9 Prctice Book A Eercises. Copy ech of the following sttements nd fill in the missing numbers: () ( ) (f) ( ) (e) (g) (h). Copy ech of the following sttements nd fill in the missing numbers: () b b b ( ) b b b b b 9 (e) ( z ) z (f) q q q. Eplin why 9.. Clculte: 0 0 () Copy ech of the following sttements nd fill in the missing numbers: () ( z ) z ( ) (f) p p p 9 9 (e) (g) ( p ) 0 p (h) q q q
8 . MEP Y9 Prctice Book A. Clculte: () (e) 0 0 (f). Fill in the missing numbers in ech of the following epressions: () 9 (e) (f) (g) (h). Fill in the missing numbers in ech of the following epressions: () 9 9. Is ech of the following sttements true or flse? () 0 0
9 MEP Y9 Prctice Book A 0. Copy nd complete ech epression: ( ) ( ) () ( ) (e) ( ) ( ) (f) 9 ( ) ( ). Negtive Indices Using negtive indices produces frctions. In this section we prctice working with negtive indices. From our work in the lst section, we see tht but we know tht, frction. So clerly, In sme wy,
10 . MEP Y9 Prctice Book A nd, in generl, n n for positive integer vlues of n. The three rules t the strt of section. cn now be used for ny integers m nd n, not just for positive vlues. Emple Clculte, leving your nswers s frctions: () () 9 Emple Simplify: () 0 9 ( ) () 9 9 ( ) +
11 MEP Y9 Prctice Book A 0 ( ) Eercises. Write the following numbers s frctions without using ny indices: () 0 (e) (f). Copy the following epressions nd fill in the missing numbers: () (e) (f) Clculte: () (e) 0 (f) +. Simplify the following epressions giving your nswers in the form of number to power: () ( ) ( ) (f) (e) 9 (g) (h) 9 9
12 . MEP Y9 Prctice Book A. Copy ech of the following epressions nd fill in the missing numbers; () (e) (f) 0. Simplify the following epressions: () 9 ( ) (e) (f) ( ). Copy nd complete the following sttements: () (e) (f) Copy the following epressions nd fill in the missing numbers: () 9 (e) ( ) (f)
13 MEP Y9 Prctice Book A 9. Copy the following epressions nd fill in the missing numbers: () If b nd b, epress s power of c, without hving ny c frctions in your finl nswer.. Stndrd Form Stndrd form is convenient wy of writing very lrge or very smll numbers. It is used on scientific clcultor when number is too lrge or too smll to be displyed on the screen. Before using stndrd form, we revise multiplying nd dividing by powers of 0. Emple Clculte: () ()
14 . MEP Y9 Prctice Book A These emples led to the pproch used for stndrd form, which is reversl of the pproch used in Emple. In stndrd form, numbers re written s 0 where < 0 nd n is n integer. n Emple Write the following numbers in stndrd form: () (e) 0.09 (f) () (e)
15 MEP Y9 Prctice Book A (f) Emple Clculte: ( ) ( ) () 0 0 ( ) ( ) ( ) + ( ) ( ) ( ) ( ) ( 0 0 ) () ( ) ( ) 0 0 ( ) ( ) ( ) + ( ) Note on Using Clcultors Your clcultor will hve key EE or EXP for entering numbers in stndrd form. For emple, for. 0, press. EXP
16 . MEP Y9 Prctice Book A which will pper on your disply like this:. 0 Some clcultors lso disply the ' 0 ' prt of the number, but not ll do. You need to find out wht your clcultor displys. Remember, you must lwys write the ' 0 ' prt when you re sked to give n nswer in stndrd form. Eercises. Clculte: () (e) 0 (f). 0 (g) 0 (h) 9. 0 (i). 0. Write ech of the following numbers in stndrd form: () (e) (f) (g) (h) Convert ech of the following numbers from stndrd form to the norml deciml nottion: () (e). 0 (f). 0 (g). 0 (h). 0 (i) Write ech of the following numbers in stndrd form: () (e) (f) (g) (h)
17 MEP Y9 Prctice Book A. Convert the following numbers from stndrd form to the norml deciml formt: () 0 0 (g). 0 0 (e). 0 (h) (f) 9. 0 (i) Without using clcultor, determine: ( ) ( ) ( 0 ) ( 0 ) () ( ) ( ) 0 0 ( ) ( ) (e). 0 0 ( ) ( ) ( ) ( ) (f) Without using clcultor, determine: ( ) ( ) ( 0 ) ( 0 ) () ( ) ( ) 0 0 (e) ( ) ( ) 0 0 ( ) ( ) (f) (. 0 ) ( 9 0 ). Without clcultor, determine the following, giving your nswers in both norml nd stndrd form:: ( ) + ( ) ( 0 ) + ( 9 0 ) () (e) (f) Use clcultor to determine: ( ) ( ) 0 ( ) (. 0 ) () ( ) ( ) 0 (. ) ( 0 ) ( ) ( ) (e) (f) (. 0 )
18 . MEP Y9 Prctice Book A 0. The rdius of the erth is. 0 m. Giving your nswers in stndrd form, correct to significnt figures, clculte the circumference of the erth in: () m cm mm km. Sir Isc Newton (-) ws mthemticin, physicist nd stronomer. In his work on the grvittionl force between two bodies he found tht he needed to multiply their msses together. () Work out the vlue of the mss of the Erth multiplied by the mss of the Moon. Give your nswer in stndrd form. Mss of Erth 9. 0 kg Mss of Moon. 0 kg Newton lso found tht he needed to work out the squre of the distnce between the two bodies. Work out the squre of the distnce between the Erth nd the Moon. Give your nswer in stndrd form. Distnce between Erth nd Moon 9. 0 km Newton's formul to clculte the grvittionl force (F) between two Gmm bodies is F where R G is the grvittionl constnt, m nd m re the msses of the two bodies, nd R is the distnce between them. Work out the grvittionl force (F) between the Sun nd the Erth Gmm using the formul F with informtion in the bo below. R Give your nswer in stndrd form. m m 9. 0 kg R. 0 km G. 0 0 (KS/9/M/Levels -/P)
19 MEP Y9 Prctice Book A. () Which of these sttements is true? (i) 0 is lrger number thn. (ii) 0 is the sme size s. (iii) 0 is smller number thn. Eplin your nswer. One of the numbers below hs the sme vlue s. 0. Write down the number. ( ) One of the numbers below hs the sme vlue s. 0. Write down the number ( ) ( ) cn be written more simply s Write the following vlues s simply s possible: ( ) ( ) (i) 0 0. (ii) 0 0 (KS/9/M/Tier -/P). Frctionl Indices Indices tht re frctions re used to represent squre roots, cube roots nd other roots of numbers. for emple, 9 for emple, for emple, n n
20 . MEP Y9 Prctice Book A Emple Clculte: () 000 () Eercises. Clculte: () 9 (e) 00 (f) (g) 9 (h) (i). Clculte: (). Clculte: () (e) (e) (f) (f)
21 MEP Y9 Prctice Book A. Clculte: () 9. Is ech of the following sttements true or flse: () Simplify: ( ) ( 0 ) () 9. Simplify: (). Clculte: 0 () ( )
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