Evaluate ((sqrt10^1009))/(sqrt10^1011-sqrt10^1007)(√101009)√101011−√101007without using a calculator?
2 Answers
Explanation:
Note that
a^b*a^c = a^(b+c)" "ab⋅ac=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))√101009√101011−√101007=10100921010112−1010072
color(white)((sqrt(10)^1009)/(sqrt(10)^(1011)-sqrt(10)^(1007))) = 10^(1007/2+1)/(10^(1007/2+2)-10^(1007/2+0))√101009√101011−√101007=1010072+11010072+2−1010072+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)
Same answer, different notation.
Explanation:
There is a common factor of
= (sqrt10^1007(sqrt10^2))/(sqrt10^1007(sqrt10^4-1))
= sqrt10^2/(sqrt10^4-1)
= 10/(100-1) = 10/99 = 0.bar(10)