Simplify.
#2sqrt(1/2)+2sqrt2-sqrt8#
In order to add or subtract numbers with square roots, the square roots must be the same.
Simplify #sqrt8# by prime factorization.
#2sqrt(1/2)+2sqrt2-sqrt(2xx2xx2)#
#2sqrt(1/2)+2sqrt2-sqrt(2^2xx2)#
#2sqrt(1/2)+2sqrt2-2sqrt2#
Simplify #sqrt(1/2)# to #(sqrt1)/(sqrt2)#.
#2xx(sqrt1)/(sqrt2)+2sqrt2-2sqrt2#
Simplify #sqrt1# to #1#.
#2xx1/(sqrt2)+2sqrt2-2sqrt2#
Rationalize the denominator by multiplying the numerator and denominator by #color(red)(sqrt2#.
#2xx1/(sqrt2)xxcolor(red)(sqrt2)/color(red)(sqrt2)+2sqrt2-2sqrt2#
Simplify.
#(2xxsqrt2)/(sqrt2xxsqrt2)+2sqrt2-2sqrt2#
Simplify.
#(2sqrt2)/2+2sqrt2-2sqrt2#
Cancel the #2/2#.
#(color(red)cancel(color(black)(2))sqrt2)/color(red)cancel(color(black)(2))+2sqrt2-2sqrt2#
#sqrt2+2sqrt2-2sqrt2#
Simplify.
#3sqrt2-2sqrt2#
Answer.
#sqrt2#
