If x = (4^3) (4^6) (4^6) (4^9) ........ (4^3x) =  (0.0625)^ (-54)  the value of x is
a) 7  b) 8  c) 9  d) 10
 

Asked by ahuja8087 | 13th Nov, 2015, 02:03: PM

Expert Answer:

C o n s i d e r space t h e space g i v e n space e q u a t i o n colon 4 cubed cross times 4 to the power of 6 cross times 4 to the power of 9 cross times 4 to the power of 12 cross times... cross times 4 to the power of 3 x end exponent equals open parentheses 0.0625 close parentheses to the power of negative 54 end exponent R e w r i t i n g space t h e space a b o v e space e q u a t i o n space w e space h a v e comma 4 cubed cross times 4 to the power of 6 cross times 4 to the power of 9 cross times 4 to the power of 12 cross times... cross times 4 to the power of 3 x end exponent equals open parentheses 1 over 16 close parentheses to the power of negative 54 end exponent rightwards double arrow 4 cubed cross times 4 to the power of 6 cross times 4 to the power of 9 cross times 4 to the power of 12 cross times... cross times 4 to the power of 3 x end exponent equals open parentheses 16 close parentheses to the power of 54 rightwards double arrow 4 to the power of 3 plus 6 plus 9 plus 12 plus... plus 3 x end exponent equals open parentheses 4 squared close parentheses to the power of 54 rightwards double arrow 4 to the power of 3 plus 6 plus 9 plus 12 plus... plus 3 x end exponent equals open parentheses 4 close parentheses to the power of 108 rightwards double arrow 3 plus 6 plus 9 plus 12 plus... plus 3 x equals 108 H e r e comma space a equals 3 comma space d equals 3 S subscript n equals n over 2 open parentheses 2 a plus open parentheses n minus 1 close parentheses d close parentheses rightwards double arrow 108 equals n over 2 open square brackets 2 cross times 3 plus open parentheses n minus 1 close parentheses cross times 3 close square brackets rightwards double arrow 108 cross times 2 equals n open square brackets 6 plus open parentheses n minus 1 close parentheses cross times 3 close square brackets rightwards double arrow 216 equals n open square brackets 6 plus 3 n minus 3 close square brackets rightwards double arrow 216 equals n open square brackets 3 n plus 3 close square brackets rightwards double arrow 3 n squared plus 3 n minus 216 equals 0 rightwards double arrow n squared plus n minus 72 equals 0 rightwards double arrow n squared plus 9 n minus 8 n minus 72 equals 0 rightwards double arrow n open parentheses n plus 9 close parentheses minus 8 open parentheses n plus 9 close parentheses equals 0 rightwards double arrow open parentheses n minus 8 close parentheses open parentheses n plus 9 close parentheses equals 0 rightwards double arrow open parentheses n minus 8 close parentheses equals 0 space o r space rightwards double arrow open parentheses n plus 9 close parentheses equals 0 rightwards double arrow n equals 8 space o r space n equals negative 9 space A s space n space c a n n o t space b e space n e g a t i v e comma space w e space h a v e comma space n equals 8 T h u s comma space l a s t space t e r m space i s space t subscript 8 t subscript 8 equals 3 plus open parentheses 8 minus 1 close parentheses cross times 3 space space space space equals 3 plus 7 cross times 3 space space space space equals 3 plus 21 space space space space equals 24 T h e r e f o r e comma space 3 x equals 24 rightwards double arrow x equals 24 over 3 equals 8 H e n c e space t h e space c o r r e c t space o p t i o n space i s space left parenthesis b right parenthesis

Answered by Vimala Ramamurthy | 13th Nov, 2015, 06:16: PM