Saturday, September 17, 2011

Nth power of a number : some more example

Nth power of a number : Some more examples

We know that

(x+a)^n=∑_(k=0)^n▒〖(n¦k) x^k a^(n-k) 〗

Now, we use the above theorem to find out the 8th power of 16 in the following manner,

〖16〗^(8 )=(1^8) (8×1^7×6^1 ) (28×1^6×6^2 ) (56×1^5×6^3 ) (70×1^4×6^4 ) (56×1^3×6^5 ) (28×1^2×6^6 ) (8×1^1×6^7 ) (6^8)

= (1) (48) (1008) (12096) (90720) (435456) (1306368) (2239488) (1679616)
= (1) (48) (1008) (12096) (90720) (435456) (1306368) (2239488+167961) (6)
= (1) (48) (1008) (12096) (90720) (435456) (1306368) (2407449) (6)
= (1) (48) (1008) (12096) (90720) (435456) (1306368+240744) (9) (6)
= (1) (48) (1008) (12096) (90720) (435456) (1547112) (9) (6)
= (1) (48) (1008) (12096) (90720) (435456+154711) (2) (9) (6)
= (1) (48) (1008) (12096) (90720) (590167) (2) (9) (6)
= (1) (48) (1008) (12096) (90720+59016) (7) (2) (9) (6)
= (1) (48) (1008) (12096) (149736) (7) (2) (9) (6)
= (1) (48) (1008) (12096+14973) (6) (7) (2) (9) (6)
= (1) (48) (1008) (27069) (6) (7) (2) (9) (6)
= (1) (48) (1008+2706) (9) (6) (7) (2) (9) (6)
= (1) (48) (3714) (9) (6) (7) (2) (9) (6)
= (1) (48+371) (4) (9) (6) (7) (2) (9) (6)
= (1) (419) (4) (9) (6) (7) (2) (9) (6)
= (1+41) (9) (4) (9) (6) (7) (2) (9) (6)
= (42) (9) (4) (9) (6) (7) (2) (9) (6)
= 4294967296. Hence the result.

Again, take another example.

〖38〗^(8 )=(3^8 ) ( 8×3^7×8^1 ) (28×3^6×8^2) (56×3^5×8^3) (70×3^4×8^4)(56×3^3×8^5) (28×3^2×8^6 ) (8×3^1×8^7) (8^8)

= (6561) (139968) (1306368) (6967296) (23224320) (49545216) (66060288) (50331648) (16777216)
= (6561) (139968) (1306368) (6967296) (23224320) (49545216) (66060288) (50331648+1677721) (6)
= (6561) (139968) (1306368) (6967296) (23224320) (49545216) (66060288) (52009369) (6)
= (6561) (139968) (1306368) (6967296) (23224320) (49545216) (66060288+5200936) (9) (6)
= (6561) (139968) (1306368) (6967296) (23224320) (49545216) (71261224) (9) (6)
= (6561) (139968) (1306368) (6967296) (23224320) (49545216+7126122) (4) (9) (6)
= (6561) (139968) (1306368) (6967296) (23224320) (56671338) (4) (9) (6)
= (6561) (139968) (1306368) (6967296) (23224320+5667133) (8) (4) (9) (6)
= (6561) (139968) (1306368) (6967296) (28891453) (8) (4) (9) (6)
= (6561) (139968) (1306368) (6967296+2889145) (3) (8) (4) (9) (6)
= (6561) (139968) (1306368) (9856441) (3) (8) (4) (9) (6)
= (6561) (139968) (1306368+985644) (1) (3) (8) (4) (9) (6)
= (6561) (139968) (2292012) (1) (3) (8) (4) (9) (6)
= (6561) (139968+229201) (2) (1) (3) (8) (4) (9) (6)
= (6561) (369169) (2) (1) (3) (8) (4) (9) (6)
= (6561+36916) (9) (2) (1) (3) (8) (4) (9) (6)
= (43477) (9) (2) (1) (3) (8) (4) (9) (6)
= 4347792138496. Hence the result.