Simplify and rationalize the denominator:\(\frac{1}{1+ \frac{1}{\sqrt{3}+1}}.\)
+128408 It's easiest to work just with the fraction in the denominator, first....so we have
1 / [ √3 + 1] [ multiply top/bottom by √3 - 1 ] =
[ √3 - 1] / [3 - 1] =
[ √3 - 1 ] / 2
So....on the bottom we have
1 + [ √3 - 1 ] / 2 =
[ 2 + √3 - 1 ] / 2 =
[ √3 + 1 ] / 2
So we have
1
___________ =
[ √3 + 1 ] / 2
2
_________ again, multiply top/bottom by √3 - 1
[ √3 + 1 ]
2 [ √3 - 1 ]
__________ =
3 - 1
2 [ √3 -1]
__________ =
2
√3 - 1
