Question

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

Answer 1

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

Explanation:

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

`a^b*a^c = a^(b+c)" "` and `" "(a^b)^c = a^(bc)`

So we find:

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

`color(white)((sqrt(10)^1009)/(sqrt(10)^(1011)-sqrt(10)^(1007))) = 10^(1007/2+1)/(10^(1007/2+2)-10^(1007/2+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)`

Answer 2

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)`