Which of the following is equivalent to " 3/1+5sqrt2." ? A, 3sqrt2-3/49. B, 1-5sqrt2/17. C, 3sqrt2-3. D, 3/3x+2. E, 1/3x+2.?
1 Answer
Nov 27, 2017
Explanation:
#3/(1+5sqrt2)#
#"we require to express the fraction with a rational"#
#"denominator"#
#"that is with no radical on the denominator"#
#"to achieve this multiply the numerator/denominator"#
#"by the "color(blue)"conjugate"" of "1+5sqrt2#
#"the conjugate of "1+5sqrt2" is "1color(red)(-)5sqrt2#
#"in general "a+-sqrtbtoa∓sqrtblarrcolor(blue)"conjugate"#
#"note that"#
#1+5sqrt2)(1-5sqrt2)larrcolor(blue)"expand using FOIL"#
#=1cancel(-5sqrt2)cancel(+5sqrt2)-(5sqrt2)^2#
#=1-50=-49larrcolor(blue)"a rational number"#
#"returning to fraction"#
#rArr(3(1-5sqrt2))/((1+5sqrt2)(1-5sqrt2))#
#=(3-15sqrt2)/(-49)=(15sqrt2-3)/49larr" multiply by " (-1)/(-1)#
#rArr3/(1+5sqrt2)-=(15sqrt2-3)/49#
