ഗണിതശാസ്ത്രം (കൃത്യങ്കങ്ങൾ) 9
കൃത്യങ്കങ്ങൾ (Exponents)
*ആവർത്തനഗുണിതത്തെ ലളിതമാക്കുന്ന രീതിയാണ് കൃത്യങ്കങ്ങളിൽ ഉൾപ്പെടുന്നത്. ഉദാഹരണമായി 2x 2 x 2 x 2= 2^4 ഇതിൽ 4 നെ 2 ന്റെ കൃതി എന്നു പറയുന്നു.
പ്രധാന കൃത്യങ്ക നിയമങ്ങൾ
(ii) a^-n = 1/a^n (ii)a^m
* a^n = a^(mn) (iii)(a^m)^n = a^mn (iv)a^m/a^n = a^m-n (v)a^m = a^n = M=n (vi)a m/n = n/a^m (vii) (a/b)^n = a^n/b^n (viii)a^0 = 1,a = 0 മാതൃകാചോദ്യങ്ങൾ
(1)
.(4^2)^3 = ------ (a)4^32 (b)4^9 (c )4^6 (d)4^23 ഉത്തരം (c ) (4^2)^3 = 4^2
*3 = 4^6 (2)
.(x ^25) ^½ = ---------------- (a)x^50 (b)x^12 1/2 (c )x^10 (d)x^5 ഉത്തരം (b) ((x)^25)^½ = (x)25
*^½ = x^ 25/2 = x^12 ½ [.'. a ^m/n = n a^m] (3)
.4^-3 ന്റെ വിലയെന്ത് ? (a)-64 (b)1/-64 (c )64 (d)1/64 ഉത്തരം (d) 4^-3 = ¼^3 = 1/64 [x-n = 1/x^n] (4)
.25x = ⅕^12 ആയാൽ x ന്റെ വിലയെന്ത് ? (a)⅙ (b)1/-6 (c )-6 (d)6 ഉത്തരം (c ) 25^x = ⅕^12 = (5^2)^x = 5^-12 = (5^2)^x = (5^2)^-6 = (25)^x = (25)-6 = x = -6 (5)
.(x^m)^n/(x^n)^m = -------------- (a)0 (b)1 (c )x^2mn (d)m/n ഉത്തരം (b) (x^m)^n/(x^n)^m = x^mn/x^mn = x^mn-mn = x^0 = 1 (6)
.2/(343)^-2 (a)1/49 (b)25/16 (c )64/49 (d)64/17 ഉത്തരം (a) 3/(343)^-2 = [(343)^-2]^⅓ =(343)^-⅔ =[(343)⅓]^-2 = (7)^-2 = 1/7^2 =1/49 [ … x -n = 1/x-n] (7)
.(5^2)^3
* (5^2)^4 / (125)^3 (a)5^2 (b)5^5 (c )5^3 (d)5^4 ഉത്തരം (b) = 5^2
*3
*5^2
*4 / (5^3)^3 =5^6
* 5^8 / 5^9 = 5^68 / 5^9 [a^m
* a^n = a^mn] = 5^14/5^9 = 5^14 - 9 = 5^5 [a^m /a^n= a^m-n] (8).
4a = 4096 ആയാൽ 4 ^a-4 എത്ര (a)4 (b)8 (c )64 (d)16 ഉത്തരം (d) 4a = 4096 = 4^a = 4^6 = a = 6 4^a -4 = 4 ^6-4 = 4^2 = 16 (9)
. y ^⅔ 1 = 17 ആയാൽ y യുടെ വിലയെന്ത്? (a) 216 (b) 312 (c ) 64 (d) 256 ഉത്തരം (c ) y^⅔ 1 = 17 y ^⅔ = 16 (y ⅔ ) ^3/2 = (16)3/2 = (16^½)^3 = (4)^3 = 64 y യുടെ വില = 64 (10)
.60 x^7 y^4 / 4x^3 y^2/15x^3y^2 (a)x^4y^2 (b)x (c )xy (d)x^2y^2 ഉത്തരം (b) 60x^7y^4 / 4 x^3y^2 = 60x^7y^4/4x^3y^2 = 15x^4y^2 60x^7y^4 / 4x^3y^2/15x^3y^2 = 15x^4y^2/15x^3y^2 = x പരിശീലന പ്രശ്നങ്ങൾ
1.
(3^2)^3
* (33)^2 / (3)^6 = ? (a)529 (b)729 (c )81 (d)2432
2.
7a = 2401 ആയാൽ a യുടെ വിലയെന്ത് ? (a)6 (b)8 (c )4 (d)2
3.
(x^n)^m
* (x^m)^n = x ആയാൽ mn = (a)1 (b)0 (c )2 (d)½
4.
5m^4
* 6m
*3m^6 = (a)90m^11 (b)90m^10 (c )30m^11 (d)60m^11
5.
(2)^-3
* (-7)^-3 = (a)(47)^-2 (b)(-14)^-3 (c )(14)^3 (d)(14)^-6
6.
5^x = 1/125 ആയാൽ x ന്റെ വില എന്ത് ? (a)3 (b)9 (c )-3 (d)-9
7.
(x^4)^2
* x ^3/x^11 = x^a, ആയാൽ a യുടെ വിലയെന്ത് ? (a)36 (b)2 (c )1 (d)0
8.
4^½
* 8^1/3
* 2^-2 = (a)2 (b)1 (c )4 (d)8
9.
[[(⅕)^-2]^2]^-1 = ? (a)1/1025 (b)1/225 (c )1/625 (d)1/25
10.
(-1)^32/5^-1 3^-1 = ? (a)132/15 (b)15/132 (c )15/2 (d)15/8 ഉത്തരങ്ങൾ
1(b) 2(c ) 3(d) 4 (a) 5 (b) 6 (c ) 7 (d) 8 (b) 9 (c ) 10 (d)
Manglish Transcribe ↓
kruthyankangal (exponents)
*aavartthanagunithatthe lalithamaakkunna reethiyaanu kruthyankangalil ulppedunnathu. Udaaharanamaayi 2x 2 x 2 x 2= 2^4 ithil 4 ne 2 nte kruthi ennu parayunnu. pradhaana kruthyanka niyamangal
(ii) a^-n = 1/a^n (ii)a^m
* a^n = a^(mn) (iii)(a^m)^n = a^mn (iv)a^m/a^n = a^m-n (v)a^m = a^n = m=n (vi)a m/n = n/a^m (vii) (a/b)^n = a^n/b^n (viii)a^0 = 1,a = 0 maathrukaachodyangal
(1)
.(4^2)^3 = ------ (a)4^32 (b)4^9 (c )4^6 (d)4^23 uttharam (c ) (4^2)^3 = 4^2
*3 = 4^6 (2)
.(x ^25) ^½ = ---------------- (a)x^50 (b)x^12 1/2 (c )x^10 (d)x^5 uttharam (b) ((x)^25)^½ = (x)25
*^½ = x^ 25/2 = x^12 ½ [.'. A ^m/n = n a^m] (3)
. 4^-3 nte vilayenthu ? (a)-64 (b)1/-64 (c )64 (d)1/64 uttharam (d) 4^-3 = ¼^3 = 1/64 [x-n = 1/x^n] (4)
. 25x = ⅕^12 aayaal x nte vilayenthu ? (a)⅙ (b)1/-6 (c )-6 (d)6 uttharam (c ) 25^x = ⅕^12 = (5^2)^x = 5^-12 = (5^2)^x = (5^2)^-6 = (25)^x = (25)-6 = x = -6 (5)
.(x^m)^n/(x^n)^m = -------------- (a)0 (b)1 (c )x^2mn (d)m/n uttharam (b) (x^m)^n/(x^n)^m = x^mn/x^mn = x^mn-mn = x^0 = 1 (6)
. 2/(343)^-2 (a)1/49 (b)25/16 (c )64/49 (d)64/17 uttharam (a) 3/(343)^-2 = [(343)^-2]^⅓ =(343)^-⅔ =[(343)⅓]^-2 = (7)^-2 = 1/7^2 =1/49 [ … x -n = 1/x-n] (7)
.(5^2)^3
* (5^2)^4 / (125)^3 (a)5^2 (b)5^5 (c )5^3 (d)5^4 uttharam (b) = 5^2
*3
*5^2
*4 / (5^3)^3 =5^6
* 5^8 / 5^9 = 5^68 / 5^9 [a^m
* a^n = a^mn] = 5^14/5^9 = 5^14 - 9 = 5^5 [a^m /a^n= a^m-n] (8).
4a = 4096 aayaal 4 ^a-4 ethra (a)4 (b)8 (c )64 (d)16 uttharam (d) 4a = 4096 = 4^a = 4^6 = a = 6 4^a -4 = 4 ^6-4 = 4^2 = 16 (9)
. Y ^⅔ 1 = 17 aayaal y yude vilayenthu? (a) 216 (b) 312 (c ) 64 (d) 256 uttharam (c ) y^⅔ 1 = 17 y ^⅔ = 16 (y ⅔ ) ^3/2 = (16)3/2 = (16^½)^3 = (4)^3 = 64 y yude vila = 64 (10)
. 60 x^7 y^4 / 4x^3 y^2/15x^3y^2 (a)x^4y^2 (b)x (c )xy (d)x^2y^2 uttharam (b) 60x^7y^4 / 4 x^3y^2 = 60x^7y^4/4x^3y^2 = 15x^4y^2 60x^7y^4 / 4x^3y^2/15x^3y^2 = 15x^4y^2/15x^3y^2 = x parisheelana prashnangal
1.
(3^2)^3
* (33)^2 / (3)^6 = ? (a)529 (b)729 (c )81 (d)2432
2.
7a = 2401 aayaal a yude vilayenthu ? (a)6 (b)8 (c )4 (d)2
3.
(x^n)^m
* (x^m)^n = x aayaal mn = (a)1 (b)0 (c )2 (d)½
4.
5m^4
* 6m
*3m^6 = (a)90m^11 (b)90m^10 (c )30m^11 (d)60m^11
5.
(2)^-3
* (-7)^-3 = (a)(47)^-2 (b)(-14)^-3 (c )(14)^3 (d)(14)^-6
6.
5^x = 1/125 aayaal x nte vila enthu ? (a)3 (b)9 (c )-3 (d)-9
7.
(x^4)^2
* x ^3/x^11 = x^a, aayaal a yude vilayenthu ? (a)36 (b)2 (c )1 (d)0
8.
4^½
* 8^1/3
* 2^-2 = (a)2 (b)1 (c )4 (d)8
9.
[[(⅕)^-2]^2]^-1 = ? (a)1/1025 (b)1/225 (c )1/625 (d)1/25
10.
(-1)^32/5^-1 3^-1 = ? (a)132/15 (b)15/132 (c )15/2 (d)15/8 uttharangal
1(b) 2(c ) 3(d) 4 (a) 5 (b) 6 (c ) 7 (d) 8 (b) 9 (c ) 10 (d)