Evaluate ((sqrt10^1009))/(sqrt10^1011-sqrt10^1007)(101009)101011101007without using a calculator?

2 Answers
Aug 19, 2017

(sqrt(10)^1009)/(sqrt(10)^(1011)-sqrt(10)^(1007)) = 10/99 = 0.bar(10)101009101011101007=1099=0.¯¯¯¯10

Explanation:

Note that sqrt(10) = 10^(1/2)10=1012 and if a, b, c > 0a,b,c>0 then:

a^b*a^c = a^(b+c)" "abac=ab+c and " "(a^b)^c = a^(bc) (ab)c=abc

So we find:

(sqrt(10)^1009)/(sqrt(10)^(1011)-sqrt(10)^(1007)) = 10^(1009/2)/(10^(1011/2)-10^(1007/2))101009101011101007=101009210101121010072

color(white)((sqrt(10)^1009)/(sqrt(10)^(1011)-sqrt(10)^(1007))) = 10^(1007/2+1)/(10^(1007/2+2)-10^(1007/2+0))101009101011101007=1010072+11010072+21010072+0

color(white)((sqrt(10)^1009)/(sqrt(10)^(1011)-sqrt(10)^(1007))) = color(red)(cancel(color(black)(10^(1007/2))))/color(red)(cancel(color(black)(10^(1007/2))))*(10/(100-1))

color(white)((sqrt(10)^1009)/(sqrt(10)^(1011)-sqrt(10)^(1007))) = 10/99 = 0.bar(10)

Aug 19, 2017

Same answer, different notation.

Explanation:

sqrt10^1009/(sqrt10^1011-sqrt10^1007)

There is a common factor of sqrt10^1007.

= (sqrt10^1007(sqrt10^2))/(sqrt10^1007(sqrt10^4-1))

= sqrt10^2/(sqrt10^4-1)

= 10/(100-1) = 10/99 = 0.bar(10)